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Sum of series:
ln((n(n+2))/(n+1)^2)
2(x-6)^(4n)/(n(16)^n)
sin(3n)/(7n)^(1/5)
(-1)^(n-1)/ln(n+1)
Identical expressions
sin(3n)/(7n)^(one / five)
sinus of (3n) divide by (7n) to the power of (1 divide by 5)
sinus of (3n) divide by (7n) to the power of (one divide by five)
sin(3n)/(7n)(1/5)
sin3n/7n1/5
sin3n/7n^1/5
sin(3n) divide by (7n)^(1 divide by 5)

Sum of series/ sin(3n)/(7n)^(1/5)

Sum of series sin(3n)/(7n)^(1/5)

The solution

You have entered [src]

  oo          
____          
\   `         
 \    sin(3*n)
  \   --------
  /   5 _____ 
 /    \/ 7*n  
/___,         
n = 1

\sum_{n=1}^{\infty} \frac{\sin{\left(3 n \right)}}{\sqrt[5]{7 n}}

Sum(sin(3*n)/(7*n)^(1/5), (n, 1, oo))

The radius of convergence of the power series

Given number:

\frac{\sin{\left(3 n \right)}}{\sqrt[5]{7 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{7^{\frac{4}{5}} \sin{\left(3 n \right)}}{7 \sqrt[5]{n}}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\sqrt[5]{n + 1} \left|{\frac{\sin{\left(3 n \right)}}{\sin{\left(3 n + 3 \right)}}}\right|}{\sqrt[5]{n}}\right)

Let's take the limit
we find

1 = \lim_{n \to \infty}\left(\frac{\sqrt[5]{n + 1} \left|{\frac{\sin{\left(3 n \right)}}{\sin{\left(3 n + 3 \right)}}}\right|}{\sqrt[5]{n}}\right)

False

The rate of convergence of the power series

The answer [src]

  oo               
____               
\   `              
 \     4/5         
  \   7   *sin(3*n)
   )  -------------
  /        5 ___   
 /       7*\/ n    
/___,              
n = 1

\sum_{n=1}^{\infty} \frac{7^{\frac{4}{5}} \sin{\left(3 n \right)}}{7 \sqrt[5]{n}}

Sum(7^(4/5)*sin(3*n)/(7*n^(1/5)), (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
```

Other calculators

How to use it?

Identical expressions

Sum of series sin(3n)/(7n)^(1/5)

The solution

Examples of finding the sum of a series