# Can anyone remind me how to rewrite a summation?

• Eclair_de_XII
In summary, the Laplace transform of sin(t) can be derived by using the equation ##∑ar^k=\frac{a}{1-r}## and setting the limits to start from 0. This will result in the final equation of ##\frac{1}{s^2+1}##, proving that ##L[sin(t)]=\frac{1}{s^2+1}##.
Eclair_de_XII

## Homework Statement

"Given: ##sin(t)=Σ\frac{(-1)^nt^{2n+1}}{(2n+1)!}##
Prove: ##L[sin(t)]=\frac{1}{s^2+1}##."

## Homework Equations

##∑ar^k=\frac{a}{1-r}##

## The Attempt at a Solution

##L[sin(t)]=L[Σ\frac{(-1)^nt^{2n+1}}{(2n+1)!}]=L[t]-\frac{1}{3!}L[t^3]+\frac{1}{5!}L[t^5]+...+\frac{1}{k!}L[t^k]-\frac{1}{(k+2)!}L[t^{k+2}]##
##L[sin(t)]=\frac{1}{s^2}-\frac{1}{s^4}+\frac{1}{s^6}+...+\frac{1}{s^{k+1}}-\frac{1}{s^{k+3}}=\sum_{i=1}^\infty (-1)^{i+1}(\frac{1}{s^{2i}})##
##\sum_{i=1}^\infty (-1)^{i+1}(\frac{1}{s^{2i}})=\frac{-1}{1-s^2}=\frac{1}{s^2-1}##

I'm stuck on primarily the last part. How would I get from the summation above to the Laplace transform of ##sin(t)##?

Last edited:
Hold on a second... ##\sum_{i=1}^\infty (-1)^{i+1}(\frac{1}{s^{2i}})=-\sum_{i=1}^\infty 1(-\frac{1}{s^2})^i=-\frac{1}{s^{-2}+1}##

Something's still not right, here...

Last edited:
Eclair_de_XII said:

## Homework Equations

##∑ar^k=\frac{a}{1-r}##
You forgot to set the limits here.

##\sum_{k=0}^\infty ar^k=\frac{a}{1-r}##

In any case, I had to turn in the homework today, so I don't really need to solve this problem anymore.

Eclair_de_XII said:
##\sum_{k=0}^\infty ar^k=\frac{a}{1-r}##

In any case, I had to turn in the homework today, so I don't really need to solve this problem anymore.
For others stumbling on the thread: this sum starts from 0, while the one from the Laplace transform starts at 1. Taking this into account solves the problem.

@Eclair_de_XII: It is actually best not to make the first reply to your thread as it removes it from the unanswered thread list, and many helpers will miss it.

DrClaude said:
It is actually best not to make the first reply to your thread as it removes it from the unanswered thread list, and many helpers will miss it.

I see. I will keep that in mind for the future. Thank you.

## 1. What is the purpose of rewriting a summation?

Rewriting a summation can help simplify a complex expression and make it easier to solve or analyze. It can also reveal patterns or relationships between terms.

## 2. How do I know when to rewrite a summation?

If you encounter a summation with a large number of terms or a complicated expression, it may be beneficial to rewrite it. Additionally, if you notice any patterns or repetition in the terms, it may be a good idea to rewrite the summation.

## 3. What are the steps for rewriting a summation?

The first step is to identify any patterns or relationships between terms. Then, you can use properties of summations, such as the distributive property or the sum of a geometric series formula, to rewrite the expression. Finally, simplify the expression if possible.

## 4. Can rewriting a summation change its value?

No, rewriting a summation does not change its value. As long as the original and rewritten expressions are equivalent, they will have the same value.

## 5. Are there any common mistakes to avoid when rewriting a summation?

One common mistake is forgetting to adjust the limits of the summation when applying properties or simplifying the expression. It is important to keep track of the limits and make sure they are still accurate in the rewritten expression.

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