Mister Exam

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Sum of series:
6/(n^2-10n+24)
x^n/sqrt(n+1)
(n^2)(tg(pi/2)^5)
(2x-4)
Identical expressions
(two +3sqrtn)/(2n- five)
(2 plus 3 square root of n) divide by (2n minus 5)
(two plus 3 square root of n) divide by (2n minus five)
(2+3√n)/(2n-5)
2+3sqrtn/2n-5
(2+3sqrtn) divide by (2n-5)
Similar expressions
(2+3sqrtn)/(2n+5)
(2-3sqrtn)/(2n-5)

Sum of series/ (2+3sqrtn)/(2n-5)

Sum of series (2+3sqrtn)/(2n-5)

The solution

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  oo             
____             
\   `            
 \            ___
  \   2 + 3*\/ n 
  /   -----------
 /      2*n - 5  
/___,            
n = 1

$$\sum_{n=1}^{\infty} \frac{3 \sqrt{n} + 2}{2 n - 5}$$

Sum((2 + 3*sqrt(n))/(2*n - 5), (n, 1, oo))

The radius of convergence of the power series

Given number:
$$\frac{3 \sqrt{n} + 2}{2 n - 5}$$
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
$$a_{n} = \frac{3 \sqrt{n} + 2}{2 n - 5}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 \sqrt{n} + 2\right) \left|{\frac{2 n - 3}{2 n - 5}}\right|}{3 \sqrt{n + 1} + 2}\right)$$
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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Identical expressions

Similar expressions

Sum of series (2+3sqrtn)/(2n-5)

The solution

Examples of finding the sum of a series