Math 2502 Test 4 (Practice) Name: Directions: Clearly identify any indeterminate

Question

Math 2502 Test 4 (Practice) Name: Directions: Clearly identify any indeterminate forms (eg, 0" or "oo oo when L'Hôpital's Rule ("LH") is used. Problem 1. True or False... with Proof" Determine whether the following statements or false, and provide justification. If a statement is false, you may simply provide a counterexam If your justification depends on a theorem, give the name of the theorem an best you can). are true d state the theorem (as (A) If linna.-0, then ak converges. (B) If linn a 0, then ak diverges. (C) If lanl converges, then an converges. (D) If 2nan converges, then (-2)an converges.

Explanation / Answer

a)
False
Simple. Take ak = 1/k
We know that the lim k ---> inf (1/k) = 0
But 1/k is a good ol' harmonic series, which we know diverges
So, FALSE

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b)
True
If the limit of ak is non-zero,
then it means that as k ---> inf, the term ak comes to some non-zero value. This means that the next term will approach closer to the limit L and so on and so forth. This goes on for ever and thus, the sum will diverge

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c)
True.
This is the proof of absolute convergence.
If a series of positive terms |an| itself will converge, then a series containing possible pos and neg terms will definitely converge

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d)
(-2)^n an can be written as
(-1)^n * 2^n * an

Now, we know 2^n * an converges

So by alternating series test,
(-1)^n * 2^n * an also converges

TRUE

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