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Sum of series:
|(-1)^n/3^n|
sqrt(n^2+3n)-sqrt(n^2-n)
sqrt(n)/(n+3)
factorial(2*n)/4^n
Identical expressions
three - one /xi- one
3 minus 1 divide by xi minus 1
three minus one divide by xi minus one
3-1 divide by xi-1
Similar expressions
3+1/xi-1
3-1/xi+1

Sum of series 3-1/xi-1

The solution

You have entered [src]

  oo             
 ___             
 \  `            
  \   /    i    \
   )  |3 - - - 1|
  /   \    x    /
 /__,            
i = 1

$$\sum_{i=1}^{\infty} \left(\left(- \frac{i}{x} + 3\right) - 1\right)$$

Sum(3 - i/x - 1, (i, 1, oo))

The radius of convergence of the power series

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

True

False

The answer [src]

     oo
oo - --
     x

$$\infty - \frac{\infty}{x}$$

oo - oo/x

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 3-1/xi-1

The solution

Examples of finding the sum of a series