Use the result of Exercise 95 to prove that if the nonnegative series ∑n=1^∞ an converges, then

step 1 In this problem, we want to prove, so the statement states that if the non -negative series, whi

Use the result of Exercise 95 to prove that if the...

Key Concepts

Behavior of Squared Terms in Convergent Series

When dealing with a convergent nonnegative series, the terms a_n eventually become less than one. Squaring these terms results in a_n², which is even smaller than a_n. This key observation allows one to directly compare the series of squared terms with the original series, ensuring that the convergence of the original sum implies the convergence of the sum of squared terms.

Comparison Test

The comparison test is a method to determine the convergence or divergence of an infinite series by comparing it to another series whose convergence behavior is known. If, after some index, each term of one series is no greater than the corresponding term of a convergent nonnegative series, then the series under consideration also converges. This technique is used in proving that the series of squared terms converges by comparing them to the original convergent series.

Series Convergence

Series convergence refers to the property that the sequence of partial sums of an infinite series approaches a finite limit. In the context of nonnegative series, this concept is particularly straightforward because the partial sums form a monotonically increasing sequence, whose boundedness guarantees convergence by the Monotone Convergence Theorem.

Nonnegative Sequences

Nonnegative sequences have all terms greater than or equal to zero. This property is crucial because it allows the use of certain monotonicity arguments and comparison tests. When all terms are nonnegative, any operation that reduces the magnitude of the terms (such as squaring values less than one) results in a sequence that can be directly compared to the original, aiding in convergence proofs.

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