What is the Radius of Convergence for zsin(z^2)?

In summary, the conversation discusses finding the radius of convergence for the Maclaurin series of the function $zsin(z^2)$. The solution is shown to be $\infty$ by using the ratio test and the mistake of dividing by $z$ is pointed out. The conversation then shifts to discussing a different limit involving a polynomial and how it goes to infinity as the variable tends to infinity. This is shown through the use of absolute values and the understanding of limits.
  • #1
aruwin
208
0
Hello.
I need explanation on why the answer for this problem is $R=\infty$.

Here's the question and the solution.

Expand the function into maclaurin series and find the radius of convergence.
$zsin(z^2)$

Solution:
$$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$

Divide both sides by z,

$$sin(z^2)=\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$

So here's the calculation but I don't know how to get the radius of convergence. Answer is $\infty$.
 
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  • #2
aruwin said:
Hello.
I need explanation on why the answer for this problem is $R=\infty$.

Here's the question and the solution.

Expand the function into maclaurin series and find the radius of convergence.
$zsin(z^2)$

Solution:
$$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$


Yes...

Divide both sides by z,

$$sin(z^2)=\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$

WHY?! You were asked to find the MacLaurin Series for $\displaystyle \begin{align*} z\sin{ \left( z^2 \right) } \end{align*}$ WHICH YOU HAVE!

I don't know how to get the radius of convergence. Answer is $\infty$.

If the MacLaurin Series for $\displaystyle \begin{align*} \sin{(z)} \end{align*}$ is convergent for all z (it is - check with the ratio test), then so is any composition such as $\displaystyle \begin{align*} \sin{ \left( z^2 \right) } \end{align*}$ and multiplying by $\displaystyle \begin{align*} z \end{align*}$ is just multiplying a number by another number, still giving a number...
 
  • #3
Prove It said:
Yes...
WHY?! You were asked to find the MacLaurin Series for $\displaystyle \begin{align*} z\sin{ \left( z^2 \right) } \end{align*}$ WHICH YOU HAVE!
If the MacLaurin Series for $\displaystyle \begin{align*} \sin{(z)} \end{align*}$ is convergent for all z (it is - check with the ratio test), then so is any composition such as $\displaystyle \begin{align*} \sin{ \left( z^2 \right) } \end{align*}$ and multiplying by $\displaystyle \begin{align*} z \end{align*}$ is just multiplying a number by another number, still giving a number...[/size]

Oh, I divided that part with z to use it to find the radius by ratio test.
I got R= 4. What is my mistake here?
 

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  • #4
How does $\lvert -4n^2 + 10n + 6 \rvert$ go to four as $n$ tends to infinity? There's a mistake in the before-last step, I think you might have been thinking of something else and weren't paying attention :p
 
  • #5
Bacterius said:
How does $\lvert -4n^2 + 10n + 6 \rvert$ go to four as $n$ tends to infinity? There's a mistake in the before-last step, I think you might have been thinking of something else and weren't paying attention :p

Well, because the other 2 terms are getting closer to 0 while 4 doesn't? As you can see in my work, I divided all terms by $n^2$ and the only significant term left is 4. I am not sure, though.
 
  • #6
aruwin said:
Well, because the other 2 terms are getting closer to 0 while 4 doesn't? As you can see in my work, I divided all terms by $n^2$ and the only significant term left is 4. I am not sure, though.

It is true that $10n + 6$ grows much slower in comparison to $4n^2$, but that's not what the problem is asking - you're trying to find the limit of that polynomial as $n$ goes to infinity, and that limit does go to infinity. An easy way to show that it does, is to show that the expression (in absolute values) eventually increases by at least $1$ every successive $n$ (but in general, any polynomial goes to either positive or negative infinity as its variable tends to infinity).
 
  • #7
Bacterius said:
It is true that $10n + 6$ grows much slower in comparison to $4n^2$, but that's not what the problem is asking - you're trying to find the limit of that polynomial as $n$ goes to infinity, and that limit does go to infinity. An easy way to show that it does, is to show that the expression (in absolute values) eventually increases by at least $1$ every successive $n$ (but in general, any polynomial goes to either positive or negative infinity as its variable tends to infinity).

I'm sorry that I am still not quite clear with this. But how does it differ from this limit? In this one, the limit is 1/4 instead of infinity and the working is almost the same as this one.
I posted the question before on this thread:
http://mathhelpboards.com/calculus-10/how-calculate-complicated-factorial-11367.html#post53249

Is it because the simplification gives a fraction while this one has a whole number?
 
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  • #8
Bacterius said:
It is true that $10n + 6$ grows much slower in comparison to $4n^2$, but that's not what the problem is asking - you're trying to find the limit of that polynomial as $n$ goes to infinity, and that limit does go to infinity. An easy way to show that it does, is to show that the expression (in absolute values) eventually increases by at least $1$ every successive $n$ (but in general, any polynomial goes to either positive or negative infinity as its variable tends to infinity).
Oh, I get it now! I just googled about limits and learned it again. I think I understand now why the lmit is infinity. Thanks!
 

FAQ: What is the Radius of Convergence for zsin(z^2)?

What is the radius of convergence?

The radius of convergence is a mathematical concept that represents the distance from the center of a power series to the points where the series converges. It is denoted by the letter R and is a positive real number or infinity.

How is the radius of convergence calculated?

The radius of convergence can be calculated using the ratio test, which involves taking the limit of the absolute value of the quotient of consecutive terms in the power series. If this limit is less than 1, then the series converges, and the radius of convergence can be determined from the limit. If the limit is greater than 1, the series diverges, and if it is equal to 1, the test is inconclusive.

What does the radius of convergence tell us about a power series?

The radius of convergence tells us about the set of points where a power series converges. If a point is within the radius of convergence, the series will converge at that point. If a point is outside the radius of convergence, the series will diverge at that point. The radius of convergence also gives us information about the behavior of the series at the boundary points of the interval determined by the radius.

What are the limitations of the radius of convergence?

The radius of convergence is only applicable to power series, which are infinite series with terms that involve increasing powers of a variable. It cannot be used for other types of series. Additionally, the radius of convergence does not tell us anything about the behavior of the series at the boundary points of the interval, as it only gives information about the interior points.

Can the radius of convergence of a power series change?

Yes, the radius of convergence can change depending on the series. It can be affected by the coefficients of the series or the variable being raised to different powers. The radius of convergence can also change if the series is multiplied or divided by a constant or if it is composed with a function. However, the radius of convergence will always be a positive real number or infinity.

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