Solving Limits & Series: Find a_n & \Sigma_{n=1}^{\infty} a_n

[tandoorichicken]

First, a quick question about limits. Is it true that if a function "flip-flops" between positive and negative values, the limit does not exist? Say in the case of

$$\lim_{n\rightarrow\infty} (-10)^{n}$$ ?

Second, I'm having some difficulty with a problem and I'm not quite sure how to start.

)If the nth partial sum of a series $\Sigma_{n=1}^{\infty} a_n$ is $s_n = 3-2^{-n}n$, find $a_n$ and $\Sigma_{n=1}^{\infty} a_n$.

 

[Zurtex]

To the first problem, this sequence has no limit but this is not always the case for oscillating sequences, for example:

$$\lim_{n \rightarrow \infty} (-10)^{-n} = 0$$

As for the second problem, have you tried calculating $S_n - S_{n-1}$?

 

[NateTG]

First, a quick question about limits. Is it true that if a function "flip-flops" between positive and negative values, the limit does not exist? Say in the case of

$$\lim_{n\rightarrow\infty} (-10)^{n}$$ ?

No. Although the limit that you list does not exist. An example of a limit that 'flip-flops' but does exist would be
$$\lim_{n\rightarrow\infty} \left(\frac{-1}{10}\right)^{n}$$

Second, I'm having some difficulty with a problem and I'm not quite sure how to start.
If the nth partial sum of a series $\Sigma_{n=1}^{\infty} a_n$ is $s_n = 3-2^{-n}n$, find $a_n$ and $\Sigma_{n=1}^{\infty} a_n$.

Can you find the first term of the series?
Once you have that, can you find the second?

Also
$$\sum_{n=1}^{\infty} a_n = \lim_{n \rightarrow \infty} s_n$$
by definition.

P.S. This is not all that important, but if you use /sum instead of /sigma then LaTeX will automatically place the sub and superscipts in the right places.

 

[dextercioby]

No.In inline text ("itex" tags),it won't.$\sum_{k=1}^{\infty}$...

Also,not to get confused,u might use "k" as a dummy summation index.It'd be $\sum_{k=1}^{n} a_{k}$...

Daniel.