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Sum of series:
x^n/n
1/n!
nx^n!
(x-4)^n/sqrt(n)
Identical expressions
(x- four)^n/sqrt(n)
(x minus 4) to the power of n divide by square root of (n)
(x minus four) to the power of n divide by square root of (n)
(x-4)^n/√(n)
(x-4)n/sqrt(n)
x-4n/sqrtn
x-4^n/sqrtn
(x-4)^n divide by sqrt(n)
Similar expressions
(x+4)^n/sqrt(n)

Sum of series (x-4)^n/sqrt(n)

The solution

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  oo          
____          
\   `         
 \           n
  \   (x - 4) 
   )  --------
  /      ___  
 /     \/ n   
/___,         
n = 1

$$\sum_{n=1}^{\infty} \frac{\left(x - 4\right)^{n}}{\sqrt{n}}$$

Sum((x - 4)^n/sqrt(n), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\left(x - 4\right)^{n}}{\sqrt{n}}$$
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
$$a_{n} = \frac{1}{\sqrt{n}}$$
and
$$x_{0} = 4$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = 4 + \lim_{n \to \infty}\left(\frac{\sqrt{n + 1}}{\sqrt{n}}\right)$$
Let's take the limit
we find
$$R = 5$$

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 (x-4)^n/sqrt(n)

The solution

Examples of finding the sum of a series