Mister Exam

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Sum of series:
((-1)^(n+1))/(2^n)
(1000^n)/n!
sin(sqrt(n^3-1)/(n^2+1))^2
factorial(2*n+1)/factorial(3*n-1)
Identical expressions
(- one)^n+ one /(2n+ one)!
( minus 1) to the power of n plus 1 divide by (2n plus 1)!
( minus one) to the power of n plus one divide by (2n plus one)!
(-1)n+1/(2n+1)!
-1n+1/2n+1!
-1^n+1/2n+1!
(-1)^n+1 divide by (2n+1)!
Similar expressions
(-1)^n+1/(2n-1)!
(-1)^n-1/(2n+1)!
(1)^n+1/(2n+1)!

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

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

The solution

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

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

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

The radius of convergence of the power series

Given number:

\left(-1\right)^{n} + \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} = \left(-1\right)^{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(-1\right)^{n} + \frac{1}{\left(2 n + 1\right)!}}{\left(-1\right)^{n + 1} + \frac{1}{\left(2 n + 3\right)!}}}\right|

Let's take the limit
we find

True

False

The rate of convergence of the power series

The graph

Examples of finding the sum of a series

The Sum of the Power Series
```
x^n/n
```
```
(x-1)^n
```
Factorial
```
1/2^(n!)
```
```
n^2/n!
```
```
x^n/n!
```
```
k!/(n!*(n+k)!)
```
Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
```
1/n^2
```
```
1/n^4
```
```
1/n^6
```
Harmonic series
```
1/n
```
Grandi's series
```
(-1)^n
```
Alternating series
```
(-1)^(n + 1)/n
```
```
(n + 2)*(-1)^(n - 1)
```
```
(3*n - 1)/(-5)^n
```
```
(-1)^(n - 1)*n/(6*n - 5)
```
Newton–Mercator series
```
(-1)^(n + 1)/n*x^n
```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

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Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series