Mister Exam

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Sum of series:
sin1/n
sin((2n-1)-x)/(2n-1)^2
n^(1/n)-1-(ln(n))/(n)
2/(4n^2-1)
Identical expressions
two /(4n^ two - one)
2 divide by (4n squared minus 1)
two divide by (4n to the power of two minus one)
2/(4n2-1)
2/4n2-1
2/(4n²-1)
2/(4n to the power of 2-1)
2/4n^2-1
2 divide by (4n^2-1)
Similar expressions
2/(4n^2+1)

Sum of series/ 2/(4n^2-1)

Sum of series 2/(4n^2-1)

The solution

You have entered [src]

  oo          
____          
\   `         
 \       2    
  \   --------
  /      2    
 /    4*n  - 1
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{2}{4 n^{2} - 1}$$

Sum(2/(4*n^2 - 1), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{2}{4 n^{2} - 1}$$
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
$$a_{n} = \frac{2}{4 n^{2} - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(4 \left(n + 1\right)^{2} - 1\right) \left|{\frac{1}{4 n^{2} - 1}}\right|\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

$$1$$

Numerical answer [src]

1.00000000000000000000000000000

1.00000000000000000000000000000

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 2/(4n^2-1)

The solution

Examples of finding the sum of a series