Best solutions to - "Use the formula for the sum of a..."

texelaare

Answered question

2020-12-21

Use the formula for the sum of a geometric series to find the sum or state that the series diverges (enter DIV for a divergent series).

\sum_{n = 2}^{\infty} \frac{5^{n}}{12^{n}}

broliY

Skilled2020-12-22Added 97 answers

Here, the given series is

\sum_{n = 2}^{\infty} \frac{5^{n}}{12^{n}}

Let us expand the given series.

\sum_{n = 2}^{\infty} \frac{5^{n}}{12^{n}} = \frac{5^{2}}{12^{2}} + \frac{5^{3}}{12^{3}} + \frac{5^{4}}{12^{4}} + . . .

(up to

\infty)

= (\frac{5}{12})^{2} + (\frac{5}{12})^{3} + (\frac{5}{12})^{4} + . . .

We clearly see that the given series is an infinite geometric series with first term

(\frac{5}{12})^{2}

and common ratio

(\frac{5}{12})

.
Again, We know that the sum of an infinite geometric series with first term a(say) and common ratio

r (| r | \leq 1)

\frac{a}{1 - r}

So,

\sum_{n = 2}^{\infty} \frac{5^{n}}{12^{n}} = \frac{\frac{5^{2}}{12^{2}}}{1 - \frac{5}{12}}

= \frac{\frac{25}{144}}{\frac{7}{12}}

= \frac{25}{144} \times \frac{12}{7}

= \frac{25}{84}

Which is the required answer.

Jeffrey Jordon

Expert2021-12-16Added 2605 answers

Answer is given below (on video)

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