Prove that if ∑ ∣an∣ converges, then ∑ an2 converges. Is the converse true? If not, give an example that shows it is false.

Name: Prove that if ∑ ∣an∣ converges, then ∑ an2 converges. Is the converse
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Description: Prove that if ∑ ∣an∣ converges, then ∑ an2 converges. Is the converse true? If not, give an example that shows it is false.

Given that ∑an converges. This implies that limn→∞an=0. By definition of limit, for every ε>0…

Answered: Prove that if ∑ ∣an∣ converges, then ∑…

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section6.4: Graphs Of Logarithmic Functions

Problem 60SE: Prove the conjecture made in the previous exercise.

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Prove that if ∑ ∣a_n∣ converges, then ∑ a_n² converges. Is the converse true? If not, give an example that shows it is false.

Expert Solution

Step 1

Given that $\sum_{} |a_{n}|$ converges.

This implies that $\lim_{n \to \infty} |a_{n}| = 0$ .

By definition of limit,

for every $ε > 0$ there exist a $N \in ℕ$ such that $|a_{n}| < ε$ for all $n > N$ .

Take $ε = 1$ .

$|a_{n}| < 1$ for all $n > N$ .

Step 2

${|a_{n}|}^{2} < |a_{n}| < 1$ for all $n > N$ .

${a_{n}}^{2} < |a_{n}| < 1$ for all $n > N$ .

Consider $\sum_{n = 1}^{\infty} {a_{n}}^{2} = \sum_{n = 1}^{N} {a_{n}}^{2} + \sum_{n = N + 1}^{\infty} {a_{n}}^{2}$ .

Note that the first is converges. Because it has finite number of terms.

By comparison test, the second sum converges.

The sum of two convergence series is convergent.

Thus, the series $\sum_{n = 1}^{\infty} {a_{n}}^{2}$ converges.

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