Sum of series (-1)/(2n-1)!

The solution

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  oo            
 ___            
 \  `           
  \      -1     
   )  ----------
  /   (2*n - 1)!
 /__,           
n = 1

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

Sum(-1/factorial(2*n - 1), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$- \frac{1}{\left(2 n - 1\right)!}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = - \frac{1}{\left(2 n - 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(2 n + 1\right)!}{\left(2 n - 1\right)!}}\right|$$
Let's take the limit
we find

False

False

The rate of convergence of the power series

The answer [src]

-sinh(1)

$$- \sinh{\left(1 \right)}$$

-sinh(1)

Numerical answer [src]

-1.17520119364380145688238185060

-1.17520119364380145688238185060

The graph

Examples of finding the sum of a series

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```
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csc(n)^2/n^3
```
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1/n^2
```
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```
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1/n
```
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```
Alternating series
```
(-1)^(n + 1)/n
```
```
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```
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```
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```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Sum of series (-1)/(2n-1)!

The solution

Examples of finding the sum of a series