Mister Exam

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Sum of series:
ln(1-(1/n^2))
sqrt(n)*sin(pi/n^2)
e^x^2
cosnx/(n/(1/3))
Identical expressions
cosnx/(n/(one / three))
co sinus of e of nx divide by (n divide by (1 divide by 3))
co sinus of e of nx divide by (n divide by (one divide by three))
cosnx/n/1/3
cosnx divide by (n divide by (1 divide by 3))
Similar expressions
cosnx/n/(1/3)

Sum of series cosnx/(n/(1/3))

The solution

You have entered [src]

  oo          
____          
\   `         
 \    cos(n*x)
  \   --------
   )   / n \  
  /    |---|  
 /     \1/3/  
/___,         
n = 1

oo ____ \ ` \ cos(n*x) \ -------- ) / n \ / |---| / \1/3/ /___, n = 1

Sum(cos(n*x)/((n/(1/3))), (n, 1, oo))

The radius of convergence of the power series

Given number:

cos(n*x)
--------
 / n \  
 |---|  
 \1/3/

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
$$a_{n} = \frac{\cos{\left(n x \right)}}{3 n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\cos{\left(n x \right)}}{\cos{\left(x \left(n + 1\right) \right)}}}\right|}{n}\right)$$
Let's take the limit
we find

True

False

The answer [src]

  oo          
 ___          
 \  `         
  \   cos(n*x)
   )  --------
  /     3*n   
 /__,         
n = 1

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

Sum(cos(n*x)/(3*n), (n, 1, oo))

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 cosnx/(n/(1/3))

The solution

Examples of finding the sum of a series