Sum of series log5(n-i)

The solution

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  oo            
 ___            
 \  `           
  \   log(n - i)
   )  ----------
  /     log(5)  
 /__,           
i = 0

$$\sum_{i=0}^{\infty} \frac{\log{\left(- i + n \right)}}{\log{\left(5 \right)}}$$

Sum(log(n - i)/log(5), (i, 0, oo))

The radius of convergence of the power series

Given number:
$$\frac{\log{\left(- i + n \right)}}{\log{\left(5 \right)}}$$
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
$$a_{i} = \frac{\log{\left(- i + n \right)}}{\log{\left(5 \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{i \to \infty} \left|{\frac{\log{\left(- (i - n) \right)}}{\log{\left(- (i - n + 1) \right)}}}\right|$$
Let's take the limit
we find

True

False

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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Sum of series log5(n-i)

The solution

Examples of finding the sum of a series