Mister Exam

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Sum of series:
n^(1/n)
5^n
2(1/(n^2+5n+6))
((-1)^n)*((x^n)/(2n*n!))
Identical expressions
(three -sin*n)/n-lnn
(3 minus sinus of multiply by n) divide by n minus lnn
(three minus sinus of multiply by n) divide by n minus lnn
(3-sinn)/n-lnn
3-sinn/n-lnn
(3-sin*n) divide by n-lnn
Similar expressions
(3-sinn)/(n-lnn)
(3+sin*n)/n-lnn
(3-sin*n)/(n-ln*n)
(3-sin*n)/n+lnn
(3-sin*n)/n-ln*n
3-sin*n/n-ln*n

Sum of series/ (3-sin*n)/n-lnn

Sum of series (3-sin*n)/n-lnn

The solution

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  oo                       
 ___                       
 \  `                      
  \   /3 - sin(n)         \
   )  |---------- - log(n)|
  /   \    n              /
 /__,                      
n = 1

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

Sum((3 - sin(n))/n - log(n), (n, 1, oo))

The radius of convergence of the power series

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

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 (3-sin*n)/n-lnn

The solution

Examples of finding the sum of a series