Sum of series n*0,5^n

The solution

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n = 0

$$\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^{n} n$$

Sum(n*(1/2)^n, (n, 0, oo))

The radius of convergence of the power series

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

False

$$R = 0$$

The rate of convergence of the power series

The answer [src]

$$2$$

Numerical answer [src]

2.00000000000000000000000000000

2.00000000000000000000000000000

The graph

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```
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```
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```
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```
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```
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```
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```
(-1)^(n + 1)/n
```
```
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```
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```
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```
Newton–Mercator series
```
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```
Exam the series for convergence
```
(3*n - 1)/(-5)^n
```

Other calculators

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Identical expressions

Sum of series n*0,5^n

The solution

Examples of finding the sum of a series