Explain why or why not Determine whether the following stateme...

Question

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

c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

Accepted Answer

Welcome back, everyone. Suppose that limit as an approaches infinity of the nth root of the absolute value of BN equals 15th. Does the series sigma of 7 BN converge absolutely. A says yes and B says no. For this problem, let's go ahead and apply the series root test limit as an approaches infinity. Of the absolute value of we're taking 7bm and we're going to take. An exponent which is 1 divided by n acquiring the root test. What we can do from here is simply apply the properties of limits. We can write the limit as N approaches infinity. Of the absolute value of 7 raises the power of 1 divided by m. Multiplied by a limit as n approaches infinity of the absolute value of bn erases the power of 1 divided by n. We know that the latter limit is 1/5, right? According to the context of the problem, limit as an approaches infinity of the n root which is represented by an exponent when divided by m. Of the absolute value of bn is 15th. So what we want to do is simply evaluate the limit of the absolute value of 7, raise to the power of 1 divided by NSn approaches infinity. First of all, we can drop the absolute value because the absolute value of 7 is positive 7. So we can write the limit as n approaches infinity of 7, raises the power of 1 divided by n multiplied by 15th. This is equal to 1/5 multiplied by. Considering our limit, we have 7 raises the power of 1 divided by an infinitely large number. Whenever one is divided by an infinitely large number, we get the limit of 0. And 7 raises the power of 0 gives us 1. So we get the limit of 15th multiplied by 1, which is 1/5. According to the root test. If our limit is less than one, then the series converges absolutely, meaning the answer to this problem is a yes. Thank you for watching.

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

c. If lim (as k → ∞) ᵏ√|aₖ| = 1/4, then ∑ 10aₖ converges absolutely.

Key Concepts

Root Test for Series Convergence

Absolute Convergence

Effect of Multiplying Series Terms by a Constant

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