## A Generalized Solution to a Specific Type of Power Series and its Trigonometric and Hyperbolic Extensions

**Authors:** Abdalla M. Aboarab

In this paper, the power series is looked at from a different perspective.
The Summation of $n^{m}a^{-bn}$ is evaluated using a new method. An assumption
is made that the power $-bn$ is multiplied by $x$ where $ x=1$; then the series
is integrated m times in order to cancel the term $n^{m} $ out leaving the term
$a^{-bn}$. By simply taking the derivative of the result m-times, an expression
to evaluate the series arises, which include only a constant term and the
m$^{th}$ order derivative of the summation of a simple geometric series.
Further applications of the method are used to evaluate the series with the
cosine, sine, hyperbolic sine, and hyperbolic cosine. Finally, the method is
used to further simplify some of the hardest forms of series to deal with: the
series $n^{m}$sinh$^{v}(ny)$, and $n^{m}$cosh$^{v}(ny)$ and
$n^{m}$sin$^{v}(ny)$.

**Comments:** 12 Pages.

**Download:** **PDF**

### Submission history

[v1] 2018-08-23 11:47:32

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