Mister Exam

Lang:

Other calculators

How to use it?
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
ln((n(n+ two))/(n+ one)^ two)
ln((n(n plus 2)) divide by (n plus 1) squared )
ln((n(n plus two)) divide by (n plus one) to the power of two)
ln((n(n+2))/(n+1)2)
lnnn+2/n+12
ln((n(n+2))/(n+1)²)
ln((n(n+2))/(n+1) to the power of 2)
lnnn+2/n+1^2
ln((n(n+2)) divide by (n+1)^2)
Similar expressions
ln((n(n-2))/(n+1)^2)
ln((n(n+2))/(n-1)^2)

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

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

The solution

You have entered [src]

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

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

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

The radius of convergence of the power series

Given number:

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

and

x_{0} = 0

d = 0

c = 1

then

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

Let's take the limit
we find

True

False

The rate of convergence of the power series

The answer [src]

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

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

Sum(log(n*(2 + n)/(1 + 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
```

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The solution

Examples of finding the sum of a series