Question
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The th term, , of an infinite series is given by .
(a) Express , in the form , where is a constant.
(b) Find the sum of the first terms of the series, and deduce the sum of
the infinite series.

Ask by Sherman Macdonald. in Malaysia
Dec 12,2024

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Answer

(a)
(b) The sum of the infinite series is .

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Bonus Knowledge

To solve the problem step by step, let’s start with part (a).

(a) Express in the form .

Given:
We can rewrite the second term:
So substituting back, we can express as:
Thus, we find:
where .

(b) Find the sum of the first terms of the series and deduce the sum of the infinite series.

The terms of the series can be expressed as follows:
Factoring out the constant:
The inside part is a geometric series with and :
This simplifies to:
Now, to find the sum of the infinite series :
Thus, the sum of the infinite series is:

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