Question

Which series is convergent? check all that apply. sigma-summation underscript n = 1 overscript infinity endscripts startfraction 2 n over n 1 endfraction sigma-summation underscript n = 1 overscript infinity endscripts startfraction n squared minus 1 over n minus 2 endfraction sigma-summation underscript n = 1 overscript infinity endscripts (one-fifth) superscript n sigma-summation underscript n = 1 overscript infinity endscripts 3 (startfraction 1 over 10 endfraction) superscript n sigma-summation underscript n = 1 overscript infinity endscripts startfraction 1 over 10 endfraction (3) superscript n

Answer 1

These series show convergence of series which are

sigma-summation underscript n = 1 overscript infinity endscripts startfraction 2 n over n 1 endfraction

sigma-summation underscript n = 1 overscript infinity endscripts 3 (startfraction 1 over 10 endfraction)

According to the statement

we have given the some series and we have to find that the out of these series which series is convergent.

So, For this purpose we have to check the convergence of series.

So,

we know that the for the convergent of series there is a one condition and if those condition will satisfy on the series then the series is convergent and if not then series is divergent.

So, the condition for convergent is a series converges if the absolute value of the sum of any finite number of sequential terms can become arbitrary small by starting the addition from a term which is far enough. So, if this condition is not satisfied the series diverges.

When we apply this condition on all given series then only 2 series satisfy this condition which is

• sigma-summation underscript n = 1 overscript infinity endscripts startfraction 2 n over n 1 endfraction
• sigma-summation underscript n = 1 overscript infinity endscripts 3 (startfraction 1 over 10 endfraction)

So, These series show convergence of series  which are

sigma-summation underscript n = 1 overscript infinity endscripts startfraction 2 n over n 1 endfraction

sigma-summation underscript n = 1 overscript infinity endscripts 3 (startfraction 1 over 10 endfraction)

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