Mister Exam

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Sum of series:
1/n^1,5
e^-sqrt(n)/sqrt(n)
1-cos(pi/n)
2^(2*n-1)/9^n
Limit of the function:
1-1/n
Identical expressions
one - one /n
1 minus 1 divide by n
one minus one divide by n
1-1 divide by n
Similar expressions
(1/(n+i-1))-1/(n+k)
1+1/n
ln(1-1/(n^2))

Sum of series/ 1-1/n

Sum of series 1-1/n

The solution

You have entered [src]

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  \   /    1\
   )  |1 - -|
  /   \    n/
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n = 2

$$\sum_{n=2}^{\infty} \left(1 - \frac{1}{n}\right)$$

Sum(1 - 1/n, (n, 2, oo))

The radius of convergence of the power series

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

True

False

The rate of convergence of the power series

The answer [src]

oo

$$\infty$$

oo

Numerical answer

The series diverges

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 1-1/n

The solution

Examples of finding the sum of a series