Mister Exam

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Sum of series:
3i
n^2*x^n
(n^3+n+5)/(n+6)
x^(n-1)/(n!)^(1/2)
Identical expressions
one /((2n- one)(2n+ one)(2n+ three))
1 divide by ((2n minus 1)(2n plus 1)(2n plus 3))
one divide by ((2n minus one)(2n plus one)(2n plus three))
1/2n-12n+12n+3
1 divide by ((2n-1)(2n+1)(2n+3))
Similar expressions
1/((2n+1)(2n+1)(2n+3))
1/((2n-1)(2n+1)(2n-3))
1/((2n-1)(2n-1)(2n+3))
1/(2n-1)*(2n+1)*(2n+3)

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

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

The solution

You have entered [src]

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

$$\sum_{n=1}^{\infty} \frac{1}{\left(2 n - 1\right) \left(2 n + 1\right) \left(2 n + 3\right)}$$

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

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

The answer [src]

1/12

$$\frac{1}{12}$$

1/12

Numerical answer [src]

0.0833333333333333333333333333333

0.0833333333333333333333333333333

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)(2n+3))

The solution

Examples of finding the sum of a series