Use the comparison or limit comparison test to decide if the following series converge. ∑n=1∞4−sin⁡nn2+1 For each series which converges, give an approximation of its sum, together with an error estimate, as follows. First calculate the sum s5 of the first 5 terms, then estimate the "tail" ∑n=6∞an by comparing it with an appropriate improper integral or geometric series.

Question

Use the comparison or limit comparison test to decide if the following series converge.   ∑n=1∞4−sin⁡nn2+1  For each series which converges, give an approximation of its sum, together with an error estimate, as follows. First calculate the sum s5 of the first 5 terms, then estimate the &quot;tail&quot; ∑n=6∞an by comparing it with an appropriate improper integral or geometric series.

Timothy Wolff · Accepted Answer

Given series is ∑n=1∞4−sin⁡nn2+1  Since 4−sin⁡nn2+1&amp;lt;3n2 (since the maximum value of sin⁡n is 1)  And ∑n=1∞3n2 is convergent  Since ∑n=1∞4−sin⁡nn2+1&amp;lt;∑n=1∞3n2 and ∑n=1∞3n2 is convergent  By comparison test ∑n=1∞4−sin⁡nn2+1 is convergent

psor32 · Accepted Answer

Use the limit test to determine the convergence of the series:  ∑n∞4−sin⁡(n)n2+1  Recall the statement of the limit test.  The convergence of a series ∑n∞an can be determined by examining the quantity ρ=limn→∞an using the following comparisons:  If |ρ|&amp;gt;0 or if ρ is undefined, the series diverges.  If ρ=0, the limit test is inconclusive.  Check the divergence of the series by computing the limit of the summand.  Take the limit of the summand as n approaches ∞:  limn→∞4−sin⁡(n)n2+1=0  INTERMEDIATE STEPS:  Find the following limit:  limn→∞4−sin⁡(n)n2+1  A bounded function times one that approaches 0 also approaches 0.  Since 3≤4−sin⁡(n)≤5 for all n, and since limn→∞1n2+1=0, we have that limn→∞4−sin⁡(n)n2+1=0:  Answer:0  If the limit of the summand as the iterator n goes to ∞ is nonzero, then the series must diverge.  Since the limit is equal to 0, the limit test is inconclusive:  Answer: The limit test is inconclusive.

user_27qwe · Accepted Answer

The second answer is more understandable

Use the comparison or limit comparison test to decide if the

Answered question

Answer & Explanation

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