# Does the Series Involving a Function with Continuous Third Derivative Converge?

• ELog
In summary, the statement given is false and can be disproven by considering the function f(x) = x. However, if we expand f as a Taylor series, we can see that the statement may hold true for certain conditions.
ELog
This problem has been bothering me for some time. Any thoughts or insights are greatly appreciated.

Consider a function, f, with continuous third derivative on [-1,1]. Prove that the series
$\sum^{\infty}_{n=1} (nf(\frac{1}{n})-nf(-\frac{1}{n}) - 2\frac{df}{dn}(0))$ converges.

Thanks in advance for any help!

I think the statement is false. Try with f(x) = x.

You also cannot have "$df/dn$" since f is not a function of n.

Perhaps you meant
$$\sum_{n=1}^\infty\left(nf(1/n)+ nf(-1/n)- 2\frac{df}{dx}(0)\right)$$

Uhh, the statement holds true for f(x)=x (assuming df/dn is actually df/dx as HoI suggested).

@HoI: You flipped a sign (typo I presume?). For your current expression f(x)=x will not hold.

As for a proof, you might want to take the Taylor expansion of f at 0. A lot of terms will drop and you'll see fun things happening.

I found the problem in the form I presented, though the derivative was with respect to x (as HoI pointed out). If I expand f as a taylor series, I think I see the fun of which you speak.

EDIT: All we know about f is that its third derivative is continuous. Does this effect the taylor series (Is it invalid to use derivatives at 0 of order > 3 if we are unsure if they even exist?)?

Last edited:
You can't use the full Taylor series expansion (you don't even know it converges).

However, you can use Taylor up to 4 terms and use the term with the second or third derivative to define a remainder term that bounds the function result.

## 1. What does it mean for a series to converge?

Convergence of a series refers to the property of a series where the terms of the series approach a finite limit as the number of terms in the series increases. In other words, as we add more and more terms to the series, the sum of these terms gets closer and closer to a specific number.

## 2. How can I determine if a series converges or diverges?

There are several methods for determining the convergence or divergence of a series, including the comparison test, ratio test, root test, and integral test. These tests involve comparing the given series to a known series or function and examining the behavior of its terms as the number of terms increases.

## 3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series where the sum of the absolute values of its terms converges. On the other hand, conditional convergence occurs when the series converges, but the sum of the absolute values of its terms diverges. In other words, the convergence of a series depends on the order in which the terms are arranged.

## 4. Can a series converge to a negative value?

Yes, a series can converge to a negative value. In fact, the sum of a convergent series can be any real number, including positive, negative, or zero.

## 5. Is it possible for a series to converge and diverge at the same time?

No, a series cannot converge and diverge at the same time. A series either converges or diverges, but not both simultaneously. If a series satisfies the conditions for both convergence and divergence, it is considered to be indeterminate.

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