Mister Exam

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Sum of series:
(1+n)/(1+n^2)
1/(n*ln(n+1))
((n^3+3n+1)^(1/2))arcsin(2/n^3)
3/((n+2)^2-1)
Limit of the function:
(1+n)/(1+n^2)
Identical expressions
(one +n)/(one +n^ two)
(1 plus n) divide by (1 plus n squared )
(one plus n) divide by (one plus n to the power of two)
(1+n)/(1+n2)
1+n/1+n2
(1+n)/(1+n²)
(1+n)/(1+n to the power of 2)
1+n/1+n^2
(1+n) divide by (1+n^2)
Similar expressions
(1+n)/(1-n^2)
(1-n)/(1+n^2)

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

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

The solution

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  oo        
____        
\   `       
 \    1 + n 
  \   ------
  /        2
 /    1 + n 
/___,       
n = 1

$$\sum_{n=1}^{\infty} \frac{n + 1}{n^{2} + 1}$$

Sum((1 + n)/(1 + n^2), (n, 1, oo))

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

Numerical answer

The series diverges

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

Similar expressions

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

The solution

Examples of finding the sum of a series