Math_newbie
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So i want to find if this series diverges or converges but i really can't figure it out.I can't even start to prove it .Any help?Thanks!!!
Prove if series diverges or converges
- Thread starter Math_newbie
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So i want to find if this series diverges or converges but i really can't figure it out.I can't even start to prove it .Any help?Thanks!!!
D
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Can value of sin(x) be positive?
Can value of sin(x) be negative?
Does the sign of sin(x) remain constant through the domain of summation?
What does that indicate?
Please show us what you have tried and exactly where you are stuck.
Please follow the rules of posting in this forum, as enunciated at:
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Please share your work/thoughts about this problem.
Math_newbie
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Basically, this is how the problem was given to me.For the first series i can't even get started, but for the second one i thought of using D’ Alembert
quotient criteria.
pka
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Here are two considerations.
Given the series \(\sum\limits_{n = 1}^\infty {{a_n}} \), then the sequence of partial sums is \(S_K=\sum\limits_{n = 1}^K {{a_n}} \).
Now the series converges if and only it the sequence \(\left(S_n\right)\) converges.
Moreover, the series converges only if \(\left(a_n\right)\to 0\).
Given \(\sum\limits_{n = 1}^\infty {\dfrac{n}{{n + 1}}} \) we can see at once that \(\left(\dfrac{n}{n+1}\right)\to 1\ne 0\) so the series does not converge.
Given the series \(\sum\limits_{n = 1}^\infty {{a_n}} \), then the sequence of partial sums is \(S_K=\sum\limits_{n = 1}^K {{a_n}} \).
Now the series converges if and only it the sequence \(\left(S_n\right)\) converges.
Moreover, the series converges only if \(\left(a_n\right)\to 0\).
Given \(\sum\limits_{n = 1}^\infty {\dfrac{n}{{n + 1}}} \) we can see at once that \(\left(\dfrac{n}{n+1}\right)\to 1\ne 0\) so the series does not converge.