Mister Exam

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Sum of series:
200(0.98)^n-1
(3x^2-2x+3)(x+1)^x
sqrt(n)*(n/(4*n-3))^(2*n)
i2
Identical expressions
(5n- eight)^(4n+ one)/(8n)
(5n minus 8) to the power of (4n plus 1) divide by (8n)
(5n minus eight) to the power of (4n plus one) divide by (8n)
(5n-8)(4n+1)/(8n)
5n-84n+1/8n
5n-8^4n+1/8n
(5n-8)^(4n+1) divide by (8n)
Similar expressions
(5n-8)^(4n-1)/(8n)
(5n+8)^(4n+1)/(8n)

Sum of series/ (5n-8)^(4n+1)/(8n)

Sum of series (5n-8)^(4n+1)/(8n)

The solution

You have entered [src]

  oo                  
____                  
\   `                 
 \             4*n + 1
  \   (5*n - 8)       
  /   ----------------
 /          8*n       
/___,                 
n = 1

\sum_{n=1}^{\infty} \frac{\left(5 n - 8\right)^{4 n + 1}}{8 n}

Sum((5*n - 8)^(4*n + 1)/((8*n)), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{\left(5 n - 8\right)^{4 n + 1}}{8 n}

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{\left(5 n - 8\right)^{4 n + 1}}{8 n}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\left(5 n - 8\right)^{4 n + 1}}\right|}{n \left|{\left(5 n - 3\right)^{4 n + 5}}\right|}\right)

Let's take the limit
we find

False

False

The rate of convergence of the power series

The answer [src]

  oo                   
____                   
\   `                  
 \              1 + 4*n
  \   (-8 + 5*n)       
  /   -----------------
 /           8*n       
/___,                  
n = 1

\sum_{n=1}^{\infty} \frac{\left(5 n - 8\right)^{4 n + 1}}{8 n}

Sum((-8 + 5*n)^(1 + 4*n)/(8*n), (n, 1, oo))

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 (5n-8)^(4n+1)/(8n)

The solution

Examples of finding the sum of a series