Question

If the sum \[\sum_{n=1}^{infinity}\left( (1+ \sin n)/(1 + 3^n) \right) \] converges, and if S represents the sum of that series and S_6 is the sum of the first 6 terms, show that \[s - s_{6} \le 1/729\] If the sum \[\sum_{n=1}^{infinity}\left( (1+ \sin n)/(1 + 3^n) \right) \] converges, and if S represents the sum of that series and S_6 is the sum of the first 6 terms, show that \[s - s_{6} \le 1/729\] @Mathematics

Answer 1

\[s=\sum_{n=1}^{\infty}\left( (1+ \sin n)/(1 + 3^n) \right)=s_6+\sum_{n=7}^{\infty}\left( (1+ \sin n)/(1 + 3^n) \right)\]

\[s-s_6=\sum_{n=7}^{\infty}\left( (1+ \sin n)/(1 + 3^n) \right)\le\sum_{n=7}^{\infty}\left( (1+ 1)/(1 + 3^n) \right)\]

\[\le\sum_{n=7}^{\infty}\left( (2)/(3^n) \right)=\frac{1}{729}\]

Answer 2

ah thank you so much! :)
I dunno why i was so stuck on this..

Answer 3

no problem