Mister Exam

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Sum of series:
k*x+b
(2^n+3^n)/n^6
n^3-1
ln((n^2+1)/n^2)
Factor polynomial:
n^3-1
Identical expressions
n^ three - one
n cubed minus 1
n to the power of three minus one
n3-1
n³-1
n to the power of 3-1
Similar expressions
n^3+1

Sum of series/ n^3-1

Sum of series n^3-1

The solution

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  oo          
 ___          
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  \   / 3    \
  /   \n  - 1/
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n = 1

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

Sum(n^3 - 1, (n, 1, oo))

The radius of convergence of the power series

Given number:
$$n^{3} - 1$$
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
$$a_{n} = n^{3} - 1$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{n^{3} - 1}\right|}{\left(n + 1\right)^{3} - 1}\right)$$
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 n^3-1

The solution

Examples of finding the sum of a series