Mister Exam

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Sum of series:
1/(n(n+3))
(factorial(n-1)/factorial(n+1))^(n*(n+1))
factorial(n)/(n^3+n^2+4)
ln(k/(k+1))
Identical expressions
one /(log(log(n))^(log(n)))
1 divide by ( logarithm of ( logarithm of (n)) to the power of ( logarithm of (n)))
one divide by ( logarithm of ( logarithm of (n)) to the power of ( logarithm of (n)))
1/(log(log(n))(log(n)))
1/loglognlogn
1/loglogn^logn
1 divide by (log(log(n))^(log(n)))

Sum of series/ 1/(log(log(n))^(log(n)))

Sum of series 1/(log(log(n))^(log(n)))

The solution

You have entered [src]

  oo                   
____                   
\   `                  
 \            1        
  \   -----------------
  /      log(n)        
 /    log      (log(n))
/___,                  
n = 3

$$\sum_{n=3}^{\infty} \frac{1}{\log{\left(\log{\left(n \right)} \right)}^{\log{\left(n \right)}}}$$

Sum(1/(log(log(n))^log(n)), (n, 3, oo))

The radius of convergence of the power series

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

False

The rate of convergence of the power series

The answer [src]

  oo                    
 ___                    
 \  `                   
  \      -log(n)        
  /   log       (log(n))
 /__,                   
n = 3

$$\sum_{n=3}^{\infty} \log{\left(\log{\left(n \right)} \right)}^{- \log{\left(n \right)}}$$

Sum(log(log(n))^(-log(n)), (n, 3, 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

Sum of series 1/(log(log(n))^(log(n)))

The solution

Examples of finding the sum of a series