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Sum of series:
sqrt(n)*sin(pi/n^2)
-6n(6n+6)+(6n-6)(6+6n)
(n*x^n)/(n+1)
1/(n(n+1)(n+2))
Identical expressions
sqrt(n)*sin(pi/n^ two)
square root of (n) multiply by sinus of ( Pi divide by n squared )
square root of (n) multiply by sinus of ( Pi divide by n to the power of two)
√(n)*sin(pi/n^2)
sqrt(n)*sin(pi/n2)
sqrtn*sinpi/n2
sqrt(n)*sin(pi/n²)
sqrt(n)*sin(pi/n to the power of 2)
sqrt(n)sin(pi/n^2)
sqrt(n)sin(pi/n2)
sqrtnsinpi/n2
sqrtnsinpi/n^2
sqrt(n)*sin(pi divide by n^2)

Sum of series/ sqrt(n)*sin(pi/n^2)

Sum of series sqrt(n)*sin(pi/n^2)

The solution

You have entered [src]

  oo               
____               
\   `              
 \      ___    /pi\
  \   \/ n *sin|--|
  /            | 2|
 /             \n /
/___,              
n = 1

\sum_{n=1}^{\infty} \sqrt{n} \sin{\left(\frac{\pi}{n^{2}} \right)}

Sum(sqrt(n)*sin(pi/n^2), (n, 1, oo))

The radius of convergence of the power series

Given number:

\sqrt{n} \sin{\left(\frac{\pi}{n^{2}} \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} = \sqrt{n} \sin{\left(\frac{\pi}{n^{2}} \right)}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{\sqrt{n} \left|{\frac{\sin{\left(\frac{\pi}{n^{2}} \right)}}{\sin{\left(\frac{\pi}{\left(n + 1\right)^{2}} \right)}}}\right|}{\sqrt{n + 1}}\right)

Let's take the limit
we find

True

False

The rate of convergence of the power series

Numerical answer [src]

4.93872735086760375010342509187

4.93872735086760375010342509187

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 sqrt(n)*sin(pi/n^2)

The solution

Examples of finding the sum of a series