It appears that the terms of the series frac{1}{1000}+frac{1}{1001}+frac{1}{1002}+frac{1}{1003}+... are less than the corresponding terms of the conve

Dillard 2021-02-11 Answered
It appears that the terms of the series
11000+11001+11002+11003+...
are less than the corresponding terms of the convergent series
1+14+19+116+...
If the statement above is correct, then the first series converges. Is this correct? Why or why not? Make a statement about how the divergence or convergence of a series is affected by the inclusion or exclusion of the first finite number of terms.
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Expert Answer

Willie
Answered 2021-02-12 Author has 95 answers

We have given series
11000+11001+11002+11003+...
And convergent series
1+14+19+116+...
Form the given two series it is clear that the first four term of the first series are less than the corresponding terms of the second series, which is convergent. But from this we can not conclude that the first series is also convergent, as we need each nth term of the first series is less than the nth term of second series.
We know Direct comparison test:
If 0 for all n
Then,
1.If n=1bn converges, then n=1an converges.
2.If n=1an diverges, then n=1bn diverges.
Let an be the nth term of the first series and bn be the nth term of the second series. Then we get
an=1999+n and bn=1n2
Notice that \(a_n gives us
1999+n<1n2
n2n999<0
(n32.11)(n+31.11)<0 [by quadratic formula]
It follows that n2n999>0 for all n33.Therefore, 0 for all n32 and an>bn for all n33
Therefore, given two sequence does not satisfying the direct comparison test hypothesis. So, we can not conclude anything about convergence of the first series by verifying hypothesis for the first few term of two series.
Indeed, the first series is divergent( by direct comparison test with the series n=11n) as we know that n=11n is divergent and 1999+n<1n for all nN
Changing a finite number of terms in a series does not change whether or not it converges, although it may change the value of its sum if it does converge.

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Jeffrey Jordon
Answered 2021-12-16 Author has 2087 answers

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