Integrating Sums (Laplace Transform)

[rmiller70015]

Homework Statement

Using:
$$\mathcal{L}\big\{t^n\big\}=\frac{n!}{s^{n+1}}\text{for all s>0}$$
Give a formula for the Laplace transform of an arbitrary nth degree polynomial
$$p(t)=a_0+a_1t^1+a_2t^2+...+a_nt^n$$

Homework Equations

$$\mathcal{L}=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt$$

The Attempt at a Solution

I figured that the function p(t) is a sum and can be expressed as:
$$\sum_{i=0}^{n}a_it^i$$
And integration gives:
$$\lim_{b\rightarrow\infty}\int_{0}^{b}\Bigg(\sum_{i=0}^{n}a_it^i\Bigg)e^{-st}dt$$

I'm not quite sure what to do next, I don't recall if I can move the integral and limit inside of the summation or not without being arrested by the math police for violating math laws.

 

[Mark44]

Homework Statement

Using:
$$\mathcal{L}\big\{t^n\big\}=\frac{n!}{s^{n+1}}\text{for all s>0}$$
Give a formula for the Laplace transform of an arbitrary nth degree polynomial
$$p(t)=a_0+a_1t^1+a_2t^2+...+a_nt^n$$

Homework Equations

$$\mathcal{L}=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt$$

For this problem, I don't see that this equation is actually relevant. You're supposed to use the equation given at the beginning of your post, in the problem statement.
In any case, what you wrote here is roughly the defintion of the Laplace Transform. It should actually be written as
$$\mathcal{L}[p(t)]=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt$$

The Attempt at a Solution

I figured that the function p(t) is a sum and can be expressed as:
$$\sum_{i=0}^{n}a_it^i$$
And integration gives:
$$\lim_{b\rightarrow\infty}\int_{0}^{b}\Bigg(\sum_{i=0}^{n}a_it^i\Bigg)e^{-st}dt$$

Again, use the equation at the top, not the definition. The only other thing you need is that $\mathcal{L}[af(t)] = a\mathcal{L}[f(t)]$, where a is a constant. IOW, the Laplace transform of a constant times a function is the constant times the Laplace transform of a function. If you need to prove this, it's simple to do using the L.T. definition.

I'm not quite sure what to do next, I don't recall if I can move the integral and limit inside of the summation or not without being arrested by the math police for violating math laws.

 

[Ray Vickson]

Homework Statement

Using:
$$\mathcal{L}\big\{t^n\big\}=\frac{n!}{s^{n+1}}\text{for all s>0}$$
Give a formula for the Laplace transform of an arbitrary nth degree polynomial
$$p(t)=a_0+a_1t^1+a_2t^2+...+a_nt^n$$

Homework Equations

$$\mathcal{L}=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt$$

The Attempt at a Solution

I figured that the function p(t) is a sum and can be expressed as:
$$\sum_{i=0}^{n}a_it^i$$
And integration gives:
$$\lim_{b\rightarrow\infty}\int_{0}^{b}\Bigg(\sum_{i=0}^{n}a_it^i\Bigg)e^{-st}dt$$

I'm not quite sure what to do next, I don't recall if I can move the integral and limit inside of the summation or not without being arrested by the math police for violating math laws.

Of course you can: the sum has a finite number of terms, and the integral of a sum is the sum of the integrals---basic calculus 101. You can sometimes have problems with trying to interchange integration and summation, but only when the summations have infinitely many terms (i.e., are infinite series).

 

[rmiller70015]

So then I pull out the a constants from the 3

For this problem, I don't see that this equation is actually relevant. You're supposed to use the equation given at the beginning of your post, in the problem statement.
In any case, what you wrote here is roughly the defintion of the Laplace Transform. It should actually be written as
$$\mathcal{L}[p(t)]=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt$$
Again, use the equation at the top, not the definition. The only other thing you need is that $\mathcal{L}[af(t)] = a\mathcal{L}[f(t)]$, where a is a constant. IOW, the Laplace transform of a constant times a function is the constant times the Laplace transform of a function. If you need to prove this, it's simple to do using the L.T. definition.

That's the problem I was thinking I had to use the definition of a Laplace transform to answer this not just a table of transforms or in this case the one they gave us. If that's the case I just get:
$$a_n\frac{n!}{s^{n+1}}$$

 

[Mark44]

So then I pull out the a constants from the 3

That's the problem I was thinking I had to use the definition of a Laplace transform to answer this not just a table of transforms or in this case the one they gave us. If that's the case I just get:
$$a_n\frac{n!}{s^{n+1}}$$

No, that's not the answer. And I didn't suggest that you use a table of transforms. What I said was, use the equation you posted in your problem statement.

 

[rmiller70015]

What I meant was in this part of the chapter the book likes to ask us to use the integral definition of a transform but on this problem they do not specifically ask that. Also I was too hasty when writing this I guess. What I wrote down on paper was $$\sum_{i=0}^{n}a_i\frac{i!}{s^{i+1}}$$

Edit: to n not infinity

 

[Mark44]

What I meant was in this part of the chapter the book likes to ask us to use the integral definition of a transform but on this problem they do not specifically ask that.

But they specifically ask that you use the formula you wrote at the top.

Also I was too hasty when writing this I guess. What I wrote down on paper was $$\sum_{i=0}^{n}a_i\frac{i!}{s^{i+1}}$$

Looks good.

Edit: to n not infinity