Mister Exam

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Sum of series:
1/((3n-2)(3n+1))
3/(n(n+2))
(n+1)/5^n
6/9n^2+12n-5
Integral of d{x}:
e^(-n)
Graphing y =:
e^(-n)
Limit of the function:
e^(-n)
Identical expressions
e^(-n)
e to the power of ( minus n)
e(-n)
e-n
e^-n
Similar expressions
e^(n)

Sum of series/ e^(-n)

Sum of series e^(-n)

The solution

You have entered [src]

  oo     
 ___     
 \  `    
  \    -n
  /   E  
 /__,    
n = 1

$$\sum_{n=1}^{\infty} e^{- n}$$

Sum(E^(-n), (n, 1, oo))

The radius of convergence of the power series

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

False

$$R = 0$$

The rate of convergence of the power series

The answer [src]

   -1  
  e    
-------
     -1
1 - e

$$\frac{1}{e \left(1 - e^{-1}\right)}$$

exp(-1)/(1 - exp(-1))

Numerical answer [src]

0.581976706869326424385002005109

0.581976706869326424385002005109

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 e^(-n)

The solution

Examples of finding the sum of a series