Mister Exam

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Sum of series:
sqrt(n+1)/2^n
exp^(n^2)
1/(k*(k+1)*(k+2))
sin(1/n)/sqrt(n)
Identical expressions
exp^(n^ two)
exponent of to the power of (n squared )
exponent of to the power of (n to the power of two)
exp(n2)
expn2
exp^(n²)
exp to the power of (n to the power of 2)
exp^n^2

Sum of series/ exp^(n^2)

Sum of series exp^(n^2)

The solution

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n = 1

$$\sum_{n=1}^{\infty} e^{n^{2}}$$

Sum(E^(n^2), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$e^{n^{2}}$$
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
$$a_{n} = e^{n^{2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(e^{n^{2}} e^{- \left(n + 1\right)^{2}}\right)$$
Let's take the limit
we find

False

False

The rate of convergence of the power series

The answer [src]

  oo       
 ___       
 \  `      
  \    / 2\
   )   \n /
  /   e    
 /__,      
n = 1

$$\sum_{n=1}^{\infty} e^{n^{2}}$$

Sum(exp(n^2), (n, 1, oo))

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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How to use it?

Identical expressions

Sum of series exp^(n^2)

The solution

Examples of finding the sum of a series