Mister Exam

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Sum of series:
1/((4n-3)(4n+1))
sen(ix)
3^(n-1)/8^(n-1)
2^(2*n-1)/9^n
Identical expressions
(nsqrt(n))/(two n^2+n+ one)
(n square root of (n)) divide by (2n squared plus n plus 1)
(n square root of (n)) divide by (two n squared plus n plus one)
(n√(n))/(2n^2+n+1)
(nsqrt(n))/(2n2+n+1)
nsqrtn/2n2+n+1
(nsqrt(n))/(2n²+n+1)
(nsqrt(n))/(2n to the power of 2+n+1)
nsqrtn/2n^2+n+1
(nsqrt(n)) divide by (2n^2+n+1)
Similar expressions
(nsqrt(n))/(2n^2+n-1)
(nsqrt(n))/(2n^2-n+1)

Sum of series/ (nsqrt(n))/(2n^2+n+1)

Sum of series (nsqrt(n))/(2n^2+n+1)

The solution

You have entered [src]

  oo              
____              
\   `             
 \          ___   
  \     n*\/ n    
   )  ------------
  /      2        
 /    2*n  + n + 1
/___,             
n = 1

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

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

The radius of convergence of the power series

Given number:

\frac{\sqrt{n} n}{\left(2 n^{2} + n\right) + 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} = \frac{n^{\frac{3}{2}}}{2 n^{2} + n + 1}

and

x_{0} = 0

d = 0

c = 1

then

1 = \lim_{n \to \infty}\left(\frac{n^{\frac{3}{2}} \left(n + 2 \left(n + 1\right)^{2} + 2\right)}{\left(n + 1\right)^{\frac{3}{2}} \left(2 n^{2} + n + 1\right)}\right)

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

  oo              
____              
\   `             
 \         3/2    
  \       n       
   )  ------------
  /              2
 /    1 + n + 2*n 
/___,             
n = 1

\sum_{n=1}^{\infty} \frac{n^{\frac{3}{2}}}{2 n^{2} + n + 1}

Sum(n^(3/2)/(1 + n + 2*n^2), (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 (nsqrt(n))/(2n^2+n+1)

The solution

Examples of finding the sum of a series