Please show every step of the soultions. Thanks in advance! Q1) Explain why the

Question

Please show every step of the soultions. Thanks in advance!

Q1)  Explain why the integral test cannot be used to decide if the series converges or diverges.

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Q2) Does the series converge or diverge? Show and explain your result!

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Q3) Explain what is wrong with the statement:

Please show every step of the soultions. Thanks in advance! Q1) Explain why the integral test cannot be used to decide if the series converges or diverges. A convergent series Sigma an, whose terms are all positive, such that the series sigma root(an) is not convergent Sigma n+1/2n+3 Q3) Explain what is wrong with the statement: Sigma e^-n sin n Q2) Does the series converge or diverge? Show and explain your result!

Explanation / Answer

Q1) The integral of the series is equal to:

(e-x(sinx + cosx))/2 and seeing how cos and sin are unknown it is not a plausible convergence test.

furthermore, one could use the comparison test:

sinn/en =< 1/en and knowing that 1/en is a the geometric series (1/e)n we can say the series converges.

Q2) we know that 2n > n and 3 > 1 so one can say 2n + 3 > n + 1 so:

(n+1)/(2n+3) < 1 so the series converges.

furthermore, as the series approaches infinity it will equal 1/2. therefore it does converge.

Q3) This statement is false because it actually would converge. for instance. if an converges the Limit would be < 1.

Using that information; knowing L is < 1 if we took the square root of L it would still be < 1. So the series would still converge.

Hope that makes sense.

Please comment if additional help is needed.

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