Mister Exam

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How to use it?
Sum of series:
(3x^2-2x+3)(x+1)^x
sqrt(n)*(n/(4*n-3))^(2*n)
arcctg((n+1)/(n^2-3))
exp(-0.01*n)
Identical expressions
four . two * four hundred and five * four
4.2 multiply by 405 multiply by 4
four . two multiply by four hundred and five multiply by four
4.24054

Sum of series/ 4.2*405*4

Sum of series 4.24054

The solution

You have entered [src]

  oo          
 ___          
 \  `         
  \   21*405  
   )  ------*4
  /     5     
 /__,         
i = 1

$$\sum_{i=1}^{\infty} 4 \frac{21 \cdot 405}{5}$$

Sum((21*405/5)*4, (i, 1, oo))

The radius of convergence of the power series

Given number:
$$4 \frac{21 \cdot 405}{5}$$
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
$$a_{i} = 6804$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{i \to \infty} 1$$
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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How to use it?

Identical expressions

Sum of series 4.2*405*4

The solution

Examples of finding the sum of a series

Sum of series 4.24054