Mister Exam

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Sum of series:
((-1)^(n+1))/n
(-1)^n/sqrt(n)
(a)^(i-1)(b)^(k-i)((a)^k+(b)^k)
(2n+6)*tg(15/(9n+11))
Identical expressions
(two *i+ one)/n
(2 multiply by i plus 1) divide by n
(two multiply by i plus one) divide by n
(2i+1)/n
2i+1/n
(2*i+1) divide by n
Similar expressions
2i+1/n
(2i+1)/n^2
(2*i-1)/n

Sum of series/ (2*i+1)/n

Sum of series (2*i+1)/n

The solution

You have entered [src]

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

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

Sum((2*i + 1)/n, (i, 1, oo))

The radius of convergence of the power series

Given number:

\frac{2 i + 1}{n}

It is a series of species

a_{i} \left(c x - x_{0}\right)^{d i}

- power series.
The radius of convergence of a power series can be calculated by the formula:

R^{d} = \frac{x_{0} + \lim_{i \to \infty} \left|{\frac{a_{i}}{a_{i + 1}}}\right|}{c}

In this case

a_{i} = \frac{2 i + 1}{n}

and

x_{0} = 0

,

d = 0

,

c = 1

then

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

Let's take the limit
we find

True

False

The answer [src]

oo
--
n

\frac{\infty}{n}

oo/n

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
```