Determine if the following converges or diverges. If convergent, find the sum of

Question

Determine if the following converges or diverges. If convergent, find the sum of the series. converges with sum = 4 converges with sum = 1/4 converges with sum = 4/3 diverges Suppose that the total salary of all the employees at a factory in a certain city is $300,000. Of this salary, 60% is spent in the city. Of the money spent in the city, 60% is again spent in the city. If this continues indefinitely, how much total money is spent in the city, that is, how much of an effect does the original total salary have on the local economy? $288,000 $352,800 $450,000 $1,800,000 Determine all values of p for which the following series converges. p > 1 p > 0 p

Explanation / Answer

we know from an identity that n/n+1 < n+1/n+2 < n+2/n+3 .... and so on

Applying similar logic over here, we can show that k/k+3 < k+1/k+4 < k+2/k+5 .... and so on.

so, the subsequent terms keep getting larger and hence the sum would diverge.

For the second part,

we get a geometric progression, with common ratio = 0.6 (i.e 60%)

So, the sum of that infinite geometric progressionwould be

300,000 [0.6/1-0.6] = 300,000 [0.6/0.4] = 300,000 x 1.5 = 450,000

As for the last part, the series would converge as k tends to infinity.

as k is in the denominator.

If the k is in numerator the series would diverge and this would be the case only if p<0.

So, for series to converge P>0 .

Hope this helps. Please rate it ASAP. Thanks.

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