Question

question
is the following infinite sum defined?
15 - 15 + 15 - ...
select the correct answer below:
the sum is defined.
the sum is not defined.
it is impossible to tell if the sum is defined.

Explanation:

Step1: Identify the series type

This is a geometric - series with first term \(a = 15\) and common ratio \(r=- 1\).

Step2: Recall the convergence condition

For a geometric series \(\sum_{n = 0}^{\infty}a\times r^{n}\), it converges if \(|r|\lt1\).

Step3: Analyze the given series

Here, \(|r| = |-1|=1\), so the series does not converge. Since the sum of an infinite series is defined when the series converges, the sum of \(15 - 15+15-\cdots\) is not defined.

Answer:

The sum is not defined.