Mister Exam

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Sum of series:
((-1)^(n+1))/(2^n)
5/n
(-1)^n*x^n/(2*n+1)
(x^n)/(n*(n+1))
Identical expressions
(- four)*(zero . nine ^n)
( minus 4) multiply by (0.9 to the power of n)
( minus four) multiply by (zero . nine to the power of n)
(-4)*(0.9n)
-4*0.9n
(-4)(0.9^n)
(-4)(0.9n)
-40.9n
-40.9^n
Similar expressions
(4)*(0.9^n)

Sum of series/ (-4)*(0.9^n)

Sum of series (-4)*(0.9^n)

The solution

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  oo          
 ___          
 \  `         
  \          n
  /   -4*9/10 
 /__,         
n = 0

$$\sum_{n=0}^{\infty} - 4 \left(\frac{9}{10}\right)^{n}$$

Sum(-4*(9/10)^n, (n, 0, oo))

The radius of convergence of the power series

Given number:
$$- 4 \left(\frac{9}{10}\right)^{n}$$
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
$$a_{n} = -4$$
and
$$x_{0} = - \frac{9}{10}$$
,
$$d = 1$$
,
$$c = 0$$
then