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# Power series Calculator

## Get detailed solutions to your math problems with our Power series step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Solved example of power series

$\int\sin\left(x\right)\div xdx$
2

Rewrite the function $\sin\left(x\right)$ as it's representation in Maclaurin series expansion

$\int\frac{\sum_{n=0}^{\infty } x^{\left(2n+1\right)}\frac{{\left(-1\right)}^n}{\left(2n+1\right)!}}{x}dx$

Bring the denominator $x$ inside the power serie

$\int\sum_{n=0}^{\infty } \frac{x^{\left(2n+1\right)}\frac{{\left(-1\right)}^n}{\left(2n+1\right)!}}{x}dx$

Multiplying the fraction by $x^{\left(2n+1\right)}$

$\int\sum_{n=0}^{\infty } \frac{x^{\left(2n+1\right)}{\left(-1\right)}^n}{x\left(2n+1\right)!}dx$
3

Bring the denominator $x$ inside the power serie

$\int\sum_{n=0}^{\infty } \frac{x^{\left(2n+1\right)}{\left(-1\right)}^n}{x\left(2n+1\right)!}dx$

4

Simplify the fraction $\frac{x^{\left(2n+1\right)}{\left(-1\right)}^n}{x\left(2n+1\right)!}$ by $x$

$\int\sum_{n=0}^{\infty } \frac{x^{\left(2n+1-1\right)}{\left(-1\right)}^n}{\left(2n+1\right)!}dx$

We can rewrite the power series as the following

$\sum_{n=0}^{\infty } \frac{1}{\left(2n+1\right)!}\int x^{\left(2n+1-1\right)}{\left(-1\right)}^ndx$

Subtract the values $1$ and $-1$

$\sum_{n=0}^{\infty } \frac{1}{\left(2n+1\right)!}\int x^{2n}{\left(-1\right)}^ndx$
5

We can rewrite the power series as the following

$\sum_{n=0}^{\infty } \frac{1}{\left(2n+1\right)!}\int x^{2n}{\left(-1\right)}^ndx$
6

The integral of a function times a constant (${\left(-1\right)}^n$) is equal to the constant times the integral of the function

$\sum_{n=0}^{\infty } \frac{1}{\left(2n+1\right)!}{\left(-1\right)}^n\int x^{2n}dx$
7

Multiplying the fraction by ${\left(-1\right)}^n$

$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^n}{\left(2n+1\right)!}\int x^{2n}dx$
8

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2n$

$\sum_{n=0}^{\infty } \frac{{\left(-1\right)}^n}{\left(2n+1\right)!}\frac{x^{\left(2n+1\right)}}{2n+1}$
9

Multiplying fractions $\frac{{\left(-1\right)}^n}{\left(2n+1\right)!} \times \frac{x^{\left(2n+1\right)}}{2n+1}$

$\sum_{n=0}^{\infty } \frac{x^{\left(2n+1\right)}{\left(-1\right)}^n}{\left(2n+1\right)!\left(2n+1\right)}$
10

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\sum_{n=0}^{\infty } \frac{x^{\left(2n+1\right)}{\left(-1\right)}^n}{\left(2n+1\right)!\left(2n+1\right)}+C_0$

$\sum_{n=0}^{\infty } \frac{x^{\left(2n+1\right)}{\left(-1\right)}^n}{\left(2n+1\right)!\left(2n+1\right)}+C_0$