Sum of series na+b/2a

The solution

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  oo             
 ___             
 \  `            
  \   /      b  \
   )  |n*a + -*a|
  /   \      2  /
 /__,            
n = 1

$$\sum_{n=1}^{\infty} \left(a n + a \frac{b}{2}\right)$$

Sum(n*a + (b/2)*a, (n, 1, oo))

The radius of convergence of the power series

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

True

False

The answer [src]

oo*a + oo*a*b

$$\infty a b + \infty a$$

oo*a + oo*a*b

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

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Sum of series na+b/2a

The solution

Examples of finding the sum of a series