$Sum of infinite series$

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$Sum of infinite series$

If you noticed, without the summation notation, it is the present value money formula with finite time period.
PV - present value
FV - future value
i - interest rate
n - number of years
PV = FV/(1+i)^n

The present value money formula with indefinite time period is
PV = D / i
D - dividend (which is FV every year)

I'm curious if the infinite summation of FV/(1+i)^n (see the picture attached) is D / i.
Step by step process and explanation appreciated.

Sequences and Series

Samuelwhiskeyjohnsonaccount

Matchmaticians Sum of infinite series File #1

File #1 (png)

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Erdos

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Your question is vague. Do you want to compute the sum in the attached picture? What is x?

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Erdos

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Your question is vague. Do you want to compute the sum in the attached picture? What is x?

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Lets evaluates the infinot sum in the picture:
\[\sum_{i=1}^{\infty}\frac{x}{(1+i)^k}=x\sum_{i=1}^{\infty}\frac{1}{(1+i)^k}=x (\frac{\frac{1}{1+i}}{1-\frac{1}{1+i}})\]
\[=x (\frac{\frac{1}{1+i}}{\frac{i}{1+i}} )=\frac{x}{i}.\]
I am guessing that $x$ might be a typo and you meant $D$ by $x$. If so, then the infinite sum above will be equal to $D/i$ as you expected.

Here we have used the formula for the sum of infinite series
\[\sum_{k=1}^{\infty}a^k=\frac{a}{1-a} \text{for} \ \ |a|<1.\]

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$Sum of infinite series$

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