Question

Determine the convergence or divergence of the series. $$ \frac{1}{201}+\frac{1}{208}+\frac{1}{227}+\frac{1}{264}+\cdots \cdot $$

Short answer

The given series is divergent.

Step-by-step solution

Identify the pattern of the series

The given series seems to follow an arithmetic pattern in the denominator, but doesn't follow a specific common difference. The difference between the terms seems to be increasing. This series doesn't follow a simple geometric or arithmetic series pattern.

Apply the Divergence Test (n-th term test)

The Divergence Test states that if the limit of the nth term of the series as n approaches infinity is not equal to zero, then the series is divergent. In this case, since the differences between successive terms increase, the limit of the nth term as n approaches infinity would be zero. Hence, this test is inconclusive for the given series.

Apply the Comparison Test

When the Divergence Test is inconclusive, the Comparison Test can be applied. This involves comparing the series with a known series. Since the nth term of the series is more than zero and less than \( \frac{1}{n} \), and the series \( \frac{1}{n} \) is a harmonic series which is divergent, it can be concluded that the given series is also divergent.