Question
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.
(a) Express
(b) Find the sum of the first
the infinite series.
Ask by Sherman Macdonald. in Malaysia
Dec 12,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
(a)
(b) The sum of the infinite series is
.
Solution

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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: