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

In summary, the limit does not exist for flip-flopping functions between positive and negative values.
  • #1
tandoorichicken
245
0
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

[tex]\lim_{n\rightarrow\infty} (-10)^{n} [/tex] ?

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 [itex]\Sigma_{n=1}^{\infty} a_n [/itex] is [itex] s_n = 3-2^{-n}n [/itex], find [itex]a_n[/itex] and [itex]\Sigma_{n=1}^{\infty} a_n [/itex].
 
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  • #2
To the first problem, this sequence has no limit but this is not always the case for oscillating sequences, for example:

[tex]\lim_{n \rightarrow \infty} (-10)^{-n} = 0[/tex]

As for the second problem, have you tried calculating [itex]S_n - S_{n-1}[/itex]?
 
  • #3
tandoorichicken said:
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

[tex]\lim_{n\rightarrow\infty} (-10)^{n} [/tex] ?

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


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 [itex]\Sigma_{n=1}^{\infty} a_n [/itex] is [itex] s_n = 3-2^{-n}n [/itex], find [itex]a_n[/itex] and [itex]\Sigma_{n=1}^{\infty} a_n [/itex].

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

Also
[tex]\sum_{n=1}^{\infty} a_n = \lim_{n \rightarrow \infty} s_n[/tex]
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.
 
  • #4
No.In inline text ("itex" tags),it won't.[itex]\sum_{k=1}^{\infty} [/itex]...:wink:

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

Daniel.
 

1. How do I find the value of a_n in a limit or series?

To find the value of a_n in a limit or series, you can use various techniques such as algebraic manipulation, substitution, or the use of known formulas. It is important to carefully examine the given problem and choose the appropriate method to solve for a_n.

2. What is the purpose of finding the value of a_n in a limit or series?

The value of a_n in a limit or series helps us understand the behavior and convergence of the given sequence or series. It also allows us to evaluate the overall sum of the series, which can be useful in many applications in mathematics and other fields.

3. Can I use the same methods to find a_n in both limits and series?

Yes, many of the methods used to find the value of a_n in a limit can also be applied to series. However, there are also specific techniques for finding a_n in series, such as the comparison test, ratio test, and integral test.

4. How do I know if a series is convergent or divergent?

A series is convergent if its partial sums approach a finite limit as n approaches infinity. On the other hand, a series is divergent if its partial sums do not approach a finite limit as n approaches infinity. To determine the convergence or divergence of a series, various tests such as the integral test, comparison test, or alternating series test can be used.

5. Are there any shortcuts or tricks for finding the value of a_n in a limit or series?

While there are some techniques and formulas that can make finding the value of a_n easier, there are no shortcuts or tricks that work for every problem. It is important to have a strong understanding of the concepts and techniques involved in finding a_n in limits and series to solve problems accurately and efficiently.

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