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• #### How to use it?

• Sum of series:
• 1/((1+r)^n)
• exp(n)
• xi-yi
• e^(nx)
• #### Identical expressions

• one /((one +r)^n)
• 1 divide by ((1 plus r) to the power of n)
• one divide by ((one plus r) to the power of n)
• 1/((1+r)n)
• 1/1+rn
• 1/1+r^n
• 1 divide by ((1+r)^n)

• 1/((1-r)^n)

# Sum of series 1/((1+r)^n)

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### The solution

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  oo
____
\
\       1
\   --------
/          n
/    (1 + r)
/___,
n = 1         
$$\sum_{n=1}^{\infty} \frac{1}{\left(r + 1\right)^{n}}$$
Sum(1/((1 + r)^n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(r + 1\right)^{n}}$$
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
$$a_{n} = 1$$
and
$$x_{0} = - r - 1$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(- r - 1 + \lim_{n \to \infty} 1\right)$$
Let's take the limit
we find
$$\frac{1}{R} = \tilde{\infty} r$$
$$R = 0$$
/         1                  1
|-------------------  for ------- < 1
|        /      1  \      |1 + r|
|(1 + r)*|1 - -----|
|        \    1 + r/
|
<    oo
|   ___
|   \  
|    \          -n
|    /   (1 + r)         otherwise
|   /__,
\  n = 1                             
$$\begin{cases} \frac{1}{\left(1 - \frac{1}{r + 1}\right) \left(r + 1\right)} & \text{for}\: \frac{1}{\left|{r + 1}\right|} < 1 \\\sum_{n=1}^{\infty} \left(r + 1\right)^{- n} & \text{otherwise} \end{cases}$$
Piecewise((1/((1 + r)*(1 - 1/(1 + r))), 1/|1 + r| < 1), (Sum((1 + r)^(-n), (n, 1, oo)), True))