Find the first five partial sums of the given series and...

Find the first five partial sums of the given series and determine whether the series appears to be convergent or divergent. If it is convergent, find its approximate sum.
$$\sum_{n=1}^{\infty} \frac{2}{n(n+1)}$$

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Key Concepts

Infinite Series Convergence

Convergence of an infinite series occurs when the sequence of its partial sums approaches a fixed, finite value as more terms are added. This concept is crucial for establishing whether an infinite series sums to a particular number or diverges to infinity.

Approximate Sum Estimation

When an infinite convergent series has partial sums that approach a limiting value, that value is considered the sum of the series. Approximate sum estimation involves using the computed partial sums or recognizing cancellation patterns to approximate this limit.

Partial Sums

Partial sums are the cumulative totals obtained by adding the terms of an infinite series up to a certain point. They are key to analyzing how the series behaves as more terms are included and are fundamental in determining convergence or divergence.

Telescoping Series

A telescoping series is characterized by terms that cancel out in succession when the series is summed. This cancellation simplifies the computation of partial sums, often leaving only a few terms that determine the overall sum as the number of terms increases.

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