Mister Exam

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Sum of series:
x^n/(1+x^n)
(-1)^n*(2*2)^(2*n)/factorial(2*n)
24/((5-3n)(2-3n))
arctan(n+3)-arctan(n+1)
Identical expressions
one /(2n+ one)*(2n+ three)
1 divide by (2n plus 1) multiply by (2n plus 3)
one divide by (2n plus one) multiply by (2n plus three)
1/(2n+1)(2n+3)
1/2n+12n+3
1 divide by (2n+1)*(2n+3)
Similar expressions
1/(2n-1)*(2n+3)
1/(2n+1)*(2n-3)

Sum of series/ 1/(2n+1)*(2n+3)

Sum of series 1/(2n+1)*(2n+3)

The solution

You have entered [src]

  oo         
 ___         
 \  `        
  \   2*n + 3
   )  -------
  /   2*n + 1
 /__,        
n = 0

$$\sum_{n=0}^{\infty} \frac{2 n + 3}{2 n + 1}$$

Sum((2*n + 3)/(2*n + 1), (n, 0, oo))

The radius of convergence of the power series

Given number:
$$\frac{2 n + 3}{2 n + 1}$$
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
$$a_{n} = \frac{2 n + 3}{2 n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(2 n + 3\right)^{2}}{\left(2 n + 1\right) \left(2 n + 5\right)}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

oo

$$\infty$$

oo

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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Identical expressions

Similar expressions

Sum of series 1/(2n+1)*(2n+3)

The solution

Examples of finding the sum of a series