Sample Questions And Detailed Answers On Infinite Series

Here are solved sample
questions on infinite series:

1. Determine whether the series
∑ (n=1 to ∞) 1/n^2 converges or
diverges.

Solution:
We can use the p-series test to
determine whether this series
converges or diverges. The p-
series test states that if ∑ 1/n^p
converges, then ∑ 1/n also
converges for p > 1, and
diverges for p ≤ 1.

In this case, p = 2, which is
greater than 1. Therefore, we
can apply the p-series test and

,conclude that the series ∑ 1/n^2
converges.

2. Find the sum of the infinite
series ∑ (n=0 to ∞) 2^n.

,Solution:
We can use the formula for the
sum of an infinite geometric
series to find the sum of this
series. The formula states that if
a is the first term and r is the
common ratio, then the sum of
the infinite series is S = a/(1-r).

In this case, a = 1 (since 2^0 = 1)
and r = 2. Therefore,

S = 1/(1-2) = -1

So the sum of the infinite series
∑ 2^n is -1.

3. Determine whether the series
∑ (n=1 to ∞) (-1)^n/√n converges

, or diverges.

Solution:
We can use the alternating
series test to determine whether
this series converges