Mister Exam

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Sum of series:
ln(k+1/k+2)
1/((2n-1)(2n-1))
cos2n
(cos(nx)*3^n)/2^n
Identical expressions
one /((2n- one)(2n- one))
1 divide by ((2n minus 1)(2n minus 1))
one divide by ((2n minus one)(2n minus one))
1/2n-12n-1
1 divide by ((2n-1)(2n-1))
Similar expressions
1/((2n-1)*2^(n-1))
(1/((2n-1)*(2^(n-1))))
1/(2n-1)(2n-1)
1/((2n+1)(2n-1))
(1/((2n-1)*(2^n)-1))
1/((2n-1)(2n+1))

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

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

The solution

You have entered [src]

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

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

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

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

The answer [src]

  2
pi 
---
 8

$$\frac{\pi^{2}}{8}$$

pi^2/8

Numerical answer [src]

1.23370055013616982735431137498

1.23370055013616982735431137498

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-1))

The solution

Examples of finding the sum of a series