Formal Power Series - Maple Help

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Formal Power Series

The convert/FormalPowerSeries functionality was completely rewritten for Maple 2022. It offers a number of advantages over previous versions:

•	Closed-form solutions can be found in a number of cases where previous versions failed.

•	Solutions in terms of $m$ -fold hypergeometric sequences for arbitrary positive integers $m$ are now supported in more cases than before.

•	Notwithstanding the name, formal Laurent and Puiseux series (i.e., with negative or fractional exponents) can be computed as well, now in more cases than before.

•	convert/FormalPowerSeries will automatically attempt to return the series coefficients in purely real form, making the previous option makereal obsolete.

•	In a number of cases, the new code returns more compact answers than previous versions.

•	If a closed form expression for the power series coefficients cannot be found, and a recurrence relation of degree 1 or 2 exists, it will be returned instead. Previously, only linear recurrences could be computed, and would only be returned if option recurrence was specified.

•	When a recurrence relation is returned, now the initial conditions are given as well.

•	Additional options give more control over the underlying algorithm(s) used and the form of the output.

Maple 2021

Maple 2022

More closed-form solutions, notably, for sums of several terms and Puiseux solutions.

$convert ((- \frac{z}{2} + \frac{z^{3}}{6}) \arctan (z), FormalPowerSeries)$

$(- \frac{1}{2} z + \frac{1}{6} z^{3}) \arctan (z)$

(1)

$map (convert, expand ((- \frac{z}{2} + \frac{z^{3}}{6}) \arctan (z)), FormalPowerSeries)$

$\sum_{k = 0}^{\infty} (- \frac{{(−1)}^{k} z^{2 k + 2}}{4 k + 2}) + (\sum_{k = 0}^{\infty} \frac{{(−1)}^{k} z^{2 k + 4}}{12 k + 6})$

(2)

$convert ((- \frac{z}{2} + \frac{z^{3}}{6}) \arctan (z), FormalPowerSeries)$

$- \frac{z^{2}}{6} + \frac{5}{9} + (\sum_{n = 0}^{\infty} \frac{(4 n - 5) {(−1)}^{n} z^{2 n}}{3 (2 n - 1) (2 n - 3)})$

(3)

$convert (\arctan (z) + \arcsin (z), FormalPowerSeries)$

$\arctan (z) + \arcsin (z)$

(4)

$map (convert, \arctan (z) + \arcsin (z), FormalPowerSeries)$

$(\sum_{k = 0}^{\infty} \frac{{(−1)}^{k} z^{2 k + 1}}{2 k + 1}) + (\sum_{k = 0}^{\infty} \frac{(2 k)! 4^{- k} z^{2 k + 1}}{{k!}^{2} (2 k + 1)})$

(5)

$convert (\arctan (z) + \arcsin (z), FormalPowerSeries)$

$\sum_{n = 0}^{\infty} \frac{({(−1)}^{n} {n!}^{2} + (2 n)! 4^{- n}) z^{2 n + 1}}{(2 n + 1) {n!}^{2}}$

(6)

$convert (\arctan (z) + \arcsin (z), FormalPowerSeries, output = expanded)$

$(\sum_{n = 0}^{\infty} \frac{{(−1)}^{n} z^{2 n + 1}}{2 n + 1}) + (\sum_{n = 0}^{\infty} \frac{(2 n)! 4^{- n} z^{2 n + 1}}{(2 n + 1) {n!}^{2}})$

(7)

$convert (\frac{{(1 - \sqrt{1 - 4 z})}^{2} z^{2}}{4 \sqrt{1 - 4 z}}, FormalPowerSeries)$

$\frac{{(1 - \sqrt{1 - 4 z})}^{2} z^{2}}{4 \sqrt{1 - 4 z}}$

(8)

$convert (\frac{{(1 - \sqrt{1 - 4 z})}^{2} z^{2}}{4 \sqrt{1 - 4 z}}, FormalPowerSeries)$

$\sum_{n = 0}^{\infty} \frac{(2 n + 2)! (n + 2) (n + 1) z^{n + 4}}{{(n + 2)!}^{2}}$

(9)

$convert (\sqrt{\sqrt{8 z^{3} + 1} - 1}, FormalPowerSeries)$

$\sqrt{\sqrt{8 z^{3} + 1} - 1}$

(10)

$convert (\sqrt{\sqrt{8 z^{3} + 1} - 1}, FormalPowerSeries)$

$\sum_{n = 0}^{\infty} \frac{{(−1)}^{n} 2^{- n + 1} (2 n + 1) (4 n)! z^{3 n + \frac{3}{2}}}{{(2 n + 1)!}^{2}}$

(11)

Solutions in purely real form by default.

$convert (\sin (z) + z \cos (z), FormalPowerSeries)$

$\sum_{k = 0}^{\infty} (- \frac{I I^{k} (k + 1)}{2 k!} + \frac{I {(−I)}^{k} (k + 1)}{2 k!}) z^{k}$

(12)

$convert (\sin (z) + z \cos (z), FormalPowerSeries, makereal)$

$\sum_{k = 0}^{\infty} \frac{\sin (\frac{k π}{2}) (k + 1) z^{k}}{k!}$

(13)

$convert (\sin (z) + z \cos (z), FormalPowerSeries)$

$\sum_{n = 0}^{\infty} \frac{2 {(−1)}^{n} (n + 1) z^{2 n + 1}}{(2 n + 1)!}$

(14)

$convert (\ln (1 + z + z^{2} + z^{3}), FormalPowerSeries)$

$\sum_{k = 0}^{\infty} (- \frac{{(−1)}^{k + 1}}{k + 1} - \frac{I^{k + 1}}{k + 1} - \frac{{(−I)}^{k + 1}}{k + 1}) z^{k + 1}$

(15)

$convert (\ln (1 + z + z^{2} + z^{3}), FormalPowerSeries, makereal)$

$\sum_{k = 0}^{\infty} \frac{({(−1)}^{k} + 2 \sin (\frac{k π}{2})) z^{k + 1}}{k + 1}$

(16)

$convert (\ln (1 + z + z^{2} + z^{3}), FormalPowerSeries)$

$(\sum_{n = 0}^{\infty} \frac{{(−1)}^{n} z^{n + 1}}{n + 1}) + (\sum_{n = 0}^{\infty} \frac{{(−1)}^{n} z^{2 n + 2}}{n + 1})$

(17)

$convert (\frac{- z^{6} + 3 z^{2}}{- 3 z^{4} + 1}, FormalPowerSeries)$

$\frac{z^{2}}{3} + (\sum_{k = 0}^{\infty} \frac{2 \sqrt{3} ({(- \frac{I}{3} 3^{\frac{3}{4}})}^{k} {(- \frac{3^{\frac{3}{4}}}{3})}^{k} {(\frac{I}{3} 3^{\frac{3}{4}})}^{k} - {(- \frac{I}{3} 3^{\frac{3}{4}})}^{k} {(- \frac{3^{\frac{3}{4}}}{3})}^{k} {(\frac{3^{\frac{3}{4}}}{3})}^{k} + {(- \frac{I}{3} 3^{\frac{3}{4}})}^{k} {(\frac{I}{3} 3^{\frac{3}{4}})}^{k} {(\frac{3^{\frac{3}{4}}}{3})}^{k} - {(- \frac{3^{\frac{3}{4}}}{3})}^{k} {(\frac{I}{3} 3^{\frac{3}{4}})}^{k} {(\frac{3^{\frac{3}{4}}}{3})}^{k}) z^{k}}{9 {(- \frac{I}{3} 3^{\frac{3}{4}})}^{k} {(- \frac{3^{\frac{3}{4}}}{3})}^{k} {(\frac{I}{3} 3^{\frac{3}{4}})}^{k} {(\frac{3^{\frac{3}{4}}}{3})}^{k}})$

(18)

$convert (\frac{- z^{6} + 3 z^{2}}{- 3 z^{4} + 1}, FormalPowerSeries, makereal)$

$\frac{z^{2}}{3} + \sum_{k = 0}^{\infty} (- \frac{2 z^{k} 3^{\frac{k}{4} + \frac{1}{2}} (2 \cos (\frac{k π}{2}) - {(−1)}^{k} - 1)}{9})$

(19)

$convert (\frac{- z^{6} + 3 z^{2}}{- 3 z^{4} + 1}, FormalPowerSeries)$

$\frac{z^{2}}{3} + (\sum_{n = 0}^{\infty} 8 3^{n - 1} z^{4 n + 2})$

(20)

More compact answers.

$convert (\frac{1}{(q__1 - z^{2}) \cdot (q__2 - z^{3})}, FormalPowerSeries, z)$

$\sum_{k = 0}^{\infty} \frac{(- 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {(−1)}^{\frac{2}{3}} {q__1}^{3} + 2 {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{\frac{4}{3}} {(−1)}^{\frac{2}{3}} q__1 - {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {(−1)}^{\frac{2}{3}} {q__1}^{\frac{3}{2}} q__2 + {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {(−1)}^{\frac{2}{3}} {q__1}^{\frac{3}{2}} q__2 + 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {(−1)}^{\frac{1}{3}} {q__1}^{3} + {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {(−1)}^{\frac{1}{3}} {q__1}^{\frac{3}{2}} q__2 - {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {(−1)}^{\frac{1}{3}} {q__1}^{\frac{3}{2}} q__2 - 2 {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{\frac{2}{3}} {q__1}^{2} + 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {q__2}^{\frac{4}{3}} q__1 - 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{\frac{4}{3}} q__1 - 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{2} - 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{2} - 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{\frac{4}{3}} {(−1)}^{\frac{2}{3}} q__1 + 4 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{\frac{2}{3}} {(−1)}^{\frac{2}{3}} {q__1}^{2} + 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {q__2}^{\frac{2}{3}} {(−1)}^{\frac{1}{3}} {q__1}^{2} - 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{\frac{2}{3}} {(−1)}^{\frac{1}{3}} {q__1}^{2} + 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{\frac{2}{3}} {q__1}^{2} - 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {q__2}^{\frac{2}{3}} {(−1)}^{\frac{2}{3}} {q__1}^{2} - 2 {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{\frac{2}{3}} {(−1)}^{\frac{2}{3}} {q__1}^{2} + {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{2} {(−1)}^{\frac{2}{3}} + {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{2} {(−1)}^{\frac{2}{3}} - {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{2} {(−1)}^{\frac{1}{3}} - {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__2}^{2} {(−1)}^{\frac{1}{3}} + 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__1}^{\frac{3}{2}} q__2 - 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__1}^{\frac{3}{2}} q__2 + 2 {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {q__1}^{3} + 2 {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} {(\sqrt{q__1})}^{k} {(- \sqrt{q__1})}^{k} {({q__2}^{\frac{1}{3}})}^{k} {q__1}^{3}) z^{k}}{2 {({q__2}^{\frac{1}{3}})}^{k} {(- \sqrt{q__1})}^{k} (\sqrt{q__1} + {q__2}^{\frac{1}{3}}) {(\sqrt{q__1})}^{k} q__1 (- \sqrt{q__1} + {q__2}^{\frac{1}{3}}) {(- {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}})}^{k} (- \sqrt{q__1} + {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}}) (\sqrt{q__1} + {(−1)}^{\frac{1}{3}} {q__2}^{\frac{1}{3}}) {({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}})}^{k} ({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}} + \sqrt{q__1}) ({(−1)}^{\frac{2}{3}} {q__2}^{\frac{1}{3}} - \sqrt{q__1}) {({(−1)}^{\frac{1}{3}} + 1)}^{2} ({(−1)}^{\frac{1}{3}} - 1) q__2}$

(21)

$convert (\frac{1}{(q__1 - z^{2}) \cdot (q__2 - z^{3})}, FormalPowerSeries, z)$

$(\sum_{n = 0}^{\infty} \frac{(- {q__1}^{2 - \frac{n}{2}} + {(−1)}^{n} {q__1}^{2 - \frac{n}{2}} + 2 {q__2}^{- \frac{1}{3} - \frac{n}{3}} {q__1}^{\frac{5}{2}} - 2 {q__1}^{- \frac{n}{2} + \frac{1}{2}} q__2 - {(−1)}^{n} {q__1}^{- 1 - \frac{n}{2}} {q__2}^{2} + 2 {q__2}^{\frac{2}{3} - \frac{n}{3}} q__1 - {q__1}^{- 1 - \frac{n}{2}} {q__2}^{2}) z^{n}}{2 ({q__1}^{3} - {q__2}^{2}) ({q__1}^{\frac{3}{2}} + q__2)}) + (\sum_{n = 0}^{\infty} \frac{q__1 {q__2}^{- n - 1} z^{3 n}}{{q__2}^{\frac{4}{3}} + {q__2}^{\frac{2}{3}} q__1 + {q__1}^{2}}) + \sum_{n = 0}^{\infty} (- \frac{{q__2}^{- n - \frac{2}{3}} z^{3 n + 1}}{{q__2}^{\frac{4}{3}} + {q__2}^{\frac{2}{3}} q__1 + {q__1}^{2}})$

(22)

Recurrence relations returned automatically if no closed form can be found, with initial conditions.
Non-linear (degree 2) recurrences can be computed.

$convert (\arcsin {(z)}^{3}, FormalPowerSeries)$

${\arcsin (z)}^{3}$

(23)

$convert (\arcsin {(z)}^{3}, FormalPowerSeries, recurrence)$

$k^{4} a (k) - 2 (k + 1) (k + 2) (k^{2} + 2 k + 2) a (k + 2) + (k + 1) (k + 2) (k + 3) (k + 4) a (k + 4) = 0$

(24)

$convert (\arcsin {(z)}^{3}, FormalPowerSeries)$

$\sum_{n = 0}^{\infty} A (n) z^{n + 1}, RESol (\{(n^{4} + 4 n^{3} + 6 n^{2} + 4 n + 1) A (n) + (- 2 n^{4} - 18 n^{3} - 62 n^{2} - 98 n - 60) A (n + 2) + (n^{4} + 14 n^{3} + 71 n^{2} + 154 n + 120) A (n + 4) = 0\}, \{A (n)\}, \{A (0) = 0, A (1) = 0, A (2) = 1, A (3) = 0\}, INFO)$

(25)

$convert (\frac{z}{{&ExponentialE;}^{z} - 1}, FormalPowerSeries)$

$\frac{z}{{&ExponentialE;}^{z} - 1}$

(26)

$convert (\frac{z}{{&ExponentialE;}^{z} - 1}, FormalPowerSeries, recurrence)$

$\frac{z}{{&ExponentialE;}^{z} - 1}$

(27)

$convert (\frac{z}{{&ExponentialE;}^{z} - 1}, FormalPowerSeries)$

$\sum_{n = 0}^{\infty} A (n) z^{n}, RESol (A (n + 3) + \frac{A (n + 2) + (\sum_{_k = 1}^{n + 2} A (_k) A (n + 3 -_k))}{n + 4}, \{A (n)\}, \{A (0) = 1, A (1) = - \frac{1}{2}, A (2) = \frac{1}{12}\}, INFO)$

(28)

$convert (LambertW (z), FormalPowerSeries)$

$LambertW (z)$

(29)

$convert (LambertW (z), FormalPowerSeries, recurrence)$

$LambertW (z)$

(30)

$convert (LambertW (z), FormalPowerSeries)$

$\sum_{n = 0}^{\infty} A (n) z^{n}, RESol (A (n + 4) + \frac{A (n + 3) + (\sum_{_k = 1}^{n + 2} (_k + 1) A (_k + 1) A (n + 3 -_k))}{n + 3}, \{A (n)\}, \{A (0) = 0, A (1) = 1, A (2) = −1, A (3) = \frac{3}{2}\}, INFO)$

(31)

New method option (by default, all three methods are tried in sequence).

$convert (\arcsin {(z)}^{2}, FormalPowerSeries)$

$\sum_{n = 0}^{\infty} \frac{2 {n!}^{2} 4^{n} z^{2 n + 2}}{(2 n + 2)!}$

(32)

$convert (\arcsin {(z)}^{2}, FormalPowerSeries, method = hypergeometric)$

$\sum_{n = 0}^{\infty} \frac{2 {n!}^{2} 4^{n} z^{2 n + 2}}{(2 n + 2)!}$

(33)

$convert (\arcsin {(z)}^{2}, FormalPowerSeries, method = holonomic)$

$\sum_{n = 0}^{\infty} A (n) z^{n + 1}, RESol (\{(- n^{2} - 2 n - 1) A (n) + (n^{2} + 5 n + 6) A (n + 2) = 0\}, \{A (n)\}, \{A (0) = 0, A (1) = 1\}, INFO)$

(34)

$convert (\arcsin {(z)}^{2}, FormalPowerSeries, method = quadratic)$

$\sum_{n = 0}^{\infty} A (n) z^{n}, RESol (A (n + 4) - \frac{4 (n + 2) A (n + 2) + (\sum_{_k = 2}^{n} (_k + 1) A (_k + 1) (n + 3 -_k) A (n + 3 -_k)) + \sum_{_k = 2}^{n + 2} (- (_k + 1) A (_k + 1) (n + 5 -_k) A (n + 5 -_k))}{4 (n + 3)}, \{A (n)\}, \{A (0) = 0, A (1) = 0, A (2) = 1, A (3) = 0\}, INFO)$

(35)

$convert (\tan (z), FormalPowerSeries)$

$\sum_{n = 0}^{\infty} A (n) z^{n}, RESol (A (n + 3) + \frac{- 2 A (n + 1) + \sum_{_k = 1}^{n} (- 2 (_k + 1) A (_k + 1) A (n + 1 -_k))}{(n + 2) (n + 3)}, \{A (n)\}, \{A (0) = 0, A (1) = 1, A (2) = 0\}, INFO)$

(36)

$convert (\tan (z), FormalPowerSeries, method = hypergeometric)$

$\tan (z)$

(37)

$convert (\tan (z), FormalPowerSeries, method = holonomic)$

$\tan (z)$

(38)

$convert (\tan (z), FormalPowerSeries, method = quadratic)$

(39)

New output option (default: combined).

$convert ({(\sin (z) + \cos (z))}^{3}, FormalPowerSeries)$

$\sum_{n = 0}^{\infty} (- \frac{{(−1)}^{n} (9^{n} - 3) z^{2 n}}{2 (2 n)!}) + (\sum_{n = 0}^{\infty} \frac{3 {(−1)}^{n} (9^{n} + 1) z^{2 n + 1}}{2 (2 n + 1)!})$

(40)

$convert ({(\sin (z) + \cos (z))}^{3}, FormalPowerSeries, output = combined)$

$\sum_{n = 0}^{\infty} (- \frac{{(−1)}^{n} (9^{n} - 3) z^{2 n}}{2 (2 n)!}) + (\sum_{n = 0}^{\infty} \frac{3 {(−1)}^{n} (9^{n} + 1) z^{2 n + 1}}{2 (2 n + 1)!})$

(41)

$convert ({(\sin (z) + \cos (z))}^{3}, FormalPowerSeries, output = expanded)$

$\sum_{n = 0}^{\infty} (- \frac{{(−1)}^{n} 9^{n} z^{2 n}}{2 (2 n)!}) + (\sum_{n = 0}^{\infty} \frac{3 {(−1)}^{n} z^{2 n}}{2 (2 n)!}) + (\sum_{n = 0}^{\infty} \frac{3 {(−1)}^{n} 9^{n} z^{2 n + 1}}{2 (2 n + 1)!}) + (\sum_{n = 0}^{\infty} \frac{3 {(−1)}^{n} z^{2 n + 1}}{2 (2 n + 1)!})$

(42)

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