Question

87. Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| \u003c |b|, then the series ∑ (k = 1 to ∞) bᵏ converges.

Accepted Answer

Step 1: Understand the problem statement. We are given that the series $$ \sum_{k=1}^\infty a^k $$ converges and that $$ |a| \u003c |b| . We $$need to determine if this implies that the series $$ \sum_{k=1}^\infty b^k $$ also converges. Step 2: Recall the convergence criteria for geometric series. A geometric series $$ \sum_{k=1}^\infty r^k $$ converges if and only if $$ |r| \u003c 1 . If$$ $$ |r| \geq 1 $$, the series diverges. Step 3: Since $$ \sum_{k=1}^\infty a^k $$ converges, it follows that $$ |a| \u003c 1 . $$This is because the series is geometric with ratio $$ a . $$Step 4: Given $$ |a| \u003c |b| $$, and knowing $$ |a| \u003c 1 $$, it is possible that $$ |b| \geq 1 . If$$ $$ |b| \geq 1 $$, then the series $$ \sum_{k=1}^\infty b^k $$ diverges by the geometric series test. Step 5: Therefore, the statement is not necessarily true. A counterexample would be choosing $$ a = 0.5 $$ and $$ b = 0.8  ($$both less than 1, so both series converge), but if $$ b = 1.2 $$, then $$ |a| \u003c |b| $$ but $$ \sum b^k $$ diverges. Hence, the statement is false.

87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.

f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| < |b|, then the series ∑ (k = 1 to ∞) bᵏ converges.

Verified video answer for a similar problem:

Key Concepts

Convergence of Infinite Series

Geometric Series and Their Convergence

Comparison of Series Based on Term Magnitudes

87. Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| < |b|, then the series ∑ (k = 1 to ∞) bᵏ converges.