Mister Exam

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Sum of series:
t^k*(-1)^k*a^(2*k)*cos(a*x)*e^(a*z)/factorial(k)
(-1)^nx^n/n!
5/(25*n^2-5*n-6)
((3*x)^n)/n!
Identical expressions
ln(n+ one)/(n^(five / four))
ln(n plus 1) divide by (n to the power of (5 divide by 4))
ln(n plus one) divide by (n to the power of (five divide by four))
ln(n+1)/(n(5/4))
lnn+1/n5/4
lnn+1/n^5/4
ln(n+1) divide by (n^(5 divide by 4))
Similar expressions
ln(n-1)/(n^(5/4))

Sum of series/ ln(n+1)/(n^(5/4))

Sum of series ln(n+1)/(n^(5/4))

The solution

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  oo            
____            
\   `           
 \    log(n + 1)
  \   ----------
  /       5/4   
 /       n      
/___,           
n = 1

$$\sum_{n=1}^{\infty} \frac{\log{\left(n + 1 \right)}}{n^{\frac{5}{4}}}$$

Sum(log(n + 1)/n^(5/4), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{\log{\left(n + 1 \right)}}{n^{\frac{5}{4}}}$$
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
$$a_{n} = \frac{\log{\left(n + 1 \right)}}{n^{\frac{5}{4}}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right)^{\frac{5}{4}} \log{\left(n + 1 \right)}}{n^{\frac{5}{4}} \log{\left(n + 2 \right)}}\right)$$
Let's take the limit
we find

True

False

The rate of convergence of the power series

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 ln(n+1)/(n^(5/4))

The solution

Examples of finding the sum of a series