Sum of series 1+x+x2+x3+

The solution

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  oo                   
 __                    
 \ `                   
  )   (1 + x + x2 + x3)
 /_,                   
n = 1

$$\sum_{n=1}^{\infty} \left(x_{3} + \left(x_{2} + \left(x + 1\right)\right)\right)$$

Sum(1 + x + x2 + x3, (n, 1, oo))

The radius of convergence of the power series

Given number:
$$x_{3} + \left(x_{2} + \left(x + 1\right)\right)$$
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
$$a_{n} = x + x_{2} + x_{3} + 1$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find

True

False

The answer [src]

oo*(1 + x + x2 + x3)

$$\infty \left(x + x_{2} + x_{3} + 1\right)$$

oo*(1 + x + x2 + x3)

Examples of finding the sum of a series

The Sum of the Power Series
```
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```
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```
1/2^(n!)
```
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n^2/n!
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x^n/n!
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k!/(n!*(n+k)!)
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Flint Hills Series
```
csc(n)^2/n^3
```
Basel problem
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1/n^2
```
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1/n^4
```
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1/n^6
```
Harmonic series
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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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Sum of series 1+x+x2+x3+

The solution

Examples of finding the sum of a series