Question

please help with SERIES problem..

\[\sum_{n=1}^{\infty} \frac{ (-1)^\left( n+1 \right) }{ 2^n+n }\]

(a) estimate the series by evaluating \[S_2\]
i put n=1,2 so S_2
\[\frac{ 1 }{ 3 }-\frac{ 1 }{ 5 }=\frac{ 2 }{ 15 }=S_2\] (is this correct?)

(b) determine the error bound for
\[\left| S_3-\sum_{n=1}^{\infty} \frac{ (-1)^\left( n \right) }{ 2^n+n } \right|\]

Answer 1

\[\sum_{n=1}^{\infty} \frac{ (-1)^\left( n+1 \right) }{ 2^n+n }\]

(a) estimate the series by evaluating \[S_2\]
i put n=1,2 so S_2
\[\frac{ 1 }{ 3 }-\frac{ 1 }{ 5 }=\frac{ 2 }{ 15 }=S_2\] (is this correct?)

(b) determine the error bound for
\[\left| S_3-\sum_{n=1}^{\infty} \frac{ (-1)^\left( n \right) }{ 2^n+n } \right|\]

Answer 2

\[\sum_{n=1}^{\infty} \frac{ (-1)^\left( n+1 \right) }{ 2^n+n }\]

(a) estimate the series by evaluating \[S_2\]
i put n=1,2 so S_2
\[\frac{ 1 }{ 3 }-\frac{ 1 }{ 5 }=\frac{ 2 }{ 15 }=S_2\] (is this correct?)

(b) determine the error bound for
\[\left| S_3-\sum_{n=1}^{\infty} \frac{ (-1)^\left( n \right) }{ 2^n+n } \right|\]

Answer 3

(b) part please..

Answer 4

you mean we have to figure out the error from that question?

Answer 5

yes it's all related problem

Answer 6

is (a) part correct?

Answer 7

if i recall correctly, the error is bound between the average of\[\int_{n}^{\infty}f~and\int_{n+1}^{\infty}f\]

Answer 8

could you please to go over step by step.?

Answer 9

@amistre64 don't understand, please, more explanation

Answer 10

the notation goes something like this
\[S_\infty=S_n+S_{n+1}\]
the whole is the sum of the parts; if you can sum up to a finite term, the remainder can be approximated as the average of n to inf, and n+1 to inf

Answer 11

the bounds tho ... thats fuzzy

Answer 12

this might do a better job at explaining the detials :)

http://tutorial.math.lamar.edu/Classes/CalcII/EstimatingSeries.aspx

Answer 13

made me more confused.. haha

Answer 14

how do i approach first? to determine the error bound?

Answer 15

my picture seems to have gotten lost ....

Answer 16

essentally; the area under f from n+1 to infinity is the lower bound and the area under f from n to infinity is the upper bound\[\int_{n+1}^{inf}f<R_n<\int_{n}^{inf}f\]

Answer 17

you determine them by taking the limit as n to infinity of the integral of f

Answer 18

the alternating part seems a bit awkward to me, but thats the principle to what ive learned

Answer 19

ok what's the next step...??

Answer 20

examples 2 and 3 of the link i posted seem like they are most apt to fit this problem

Answer 21

\[\left|\frac{(-1)^\left( n+1 \right) }{ 2^n+n }\right|< (\frac{ 1 }{ 2 })^n\]