Answer to Question #141022 in Calculus for Besmallah Yousefi

Question #141022
Determine whether the following series converges:
S=1-(1/2)-(1/2^2)+(1/2^3)+(1/2^4)-(1/2^5)-(1/2^6)...
1+(1/3sqrt(2))+(1/3sqrt(3)+...+(1/3sqrt(k))...
1
Expert's answer
2020-11-04T12:34:55-0500

Given series:

S = 1 - "\\frac{1}{2} - \\frac{1}{2^2} + \\frac{1}{2^3}+\\frac{1}{2^4}-\\frac{1}{2^5}-\\frac{1}{2^6}+..."

Let us rearrange the given terms of the series. Then, we get

S = "[1 - \\frac{1}{2^2} + \\frac{1}{2^4} - \\frac{1}{2^6} + ... ] - [\\frac{1}{2} - \\frac{1}{2^3} + \\frac{1}{2^5} + ... ]"

Recollect the following:

"a + ar + ar^2 + ar^3 + ... + ... = \\frac{a}{1 - r}," where "a" is the first term and "r" is the common ratio of the infinite geometric series.


Using the above, we have

"S = [\\frac{1}{1+\\frac{1}{2^2}}] (a = 1, r = \\frac{-1}{2^2}) -\n [\\frac{1\/2}{1+\\frac{1}{2^2}}] (a = 1, r = \\frac{-1}{2^2})"

"= \\frac{1}{5\/4} - \\frac{1\/2} {5\/4}"

"= \\frac{4}{5} - \\frac{2}{5}"

"= \\frac{2}{5}" (finite value)


Observe that the sum of the given series is finite value.

Therefore, the given series is convergent.

Ans: Convergent


Given series: "1 + \\frac{1}{3\\sqrt2} + \\frac{1}{3\\sqrt3} + ... +\\frac{1}{3\\sqrt k} + ..."

Consider the series from the given series


"\\frac{1}{3\\sqrt2} + \\frac{1}{3\\sqrt3} + ... + \\frac{1}{3\\sqrt k} + ..."

Rewrite the above series as sigma notation.

"\\frac{1}{3\\sqrt2} + \\frac{1}{3\\sqrt3} + ... + \\frac{1}{3\\sqrt k} =\\textstyle\\sum_2^k\\frac{1}{3\\sqrt n}"

Recollect the following:

Auxiliary series "\\Sigma\\frac{1}{n^p}" is convergent series if "p" > 1 and divergent if p "\\leqslant1"

Using the above, the series "\\textstyle\\sum_2^k\\frac{1}{3\\sqrt n}" is divergent because "p = \\frac{1}{2} < 1"

We know that the convergence or divergence of a series is unaltered by adding or deleting some finite number of terms.

Using the above, the series "1 + \\textstyle\\sum_2^k\\frac{1}{3\\sqrt n}" is divergent.

Therefore, the given series is divergent.

Ans: Divergent

  





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