Max of Sum of Sines: Find the Max Value for Even n

  • Context: Graduate 
  • Thread starter Thread starter ekkilop
  • Start date Start date
  • Tags Tags
    Max Sum
Click For Summary
SUMMARY

The discussion centers on determining the maximum value of the function \(\frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}\) for even values of \(n\) within the interval \(0 \leq x \leq \pi/2\). It is established that for odd \(n\), the maximum occurs at \(x=0\) with a value of \(\sum_{k=1}^m k^n\). The challenge lies in identifying the maximum for even \(n\) and the conditions under which this occurs, particularly when \(m\) is odd.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and maxima/minima analysis.
  • Familiarity with trigonometric functions, particularly sine functions.
  • Knowledge of series summation and its properties.
  • Basic concepts of mathematical proofs and logical reasoning.
NEXT STEPS
  • Research the properties of derivatives of trigonometric functions.
  • Explore the behavior of the sine function in the context of series summation.
  • Study the implications of even and odd functions in calculus.
  • Investigate mathematical proof techniques for establishing maximum values in functions.
USEFUL FOR

Mathematicians, calculus students, and anyone interested in advanced function analysis and optimization techniques.

ekkilop
Messages
29
Reaction score
0
Hi!

Consider the function

\frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}, \quad 0 \leq x \leq \pi/2.

If n is odd this function attains its largest value, \sum_{k=1}^m k^n at x=0. But what about if n is even? Where does the maximum occur and what value does it take?

Any help is much appreciated. Thank you!
 
Last edited:
Physics news on Phys.org
Please make some attempt at the problem when you ask for help.
i.e. how would you normally go about finding the maximum?
Can you prove the statement about when m is odd?
 

Similar threads

Undergrad Understanding a proof of inexistence of max nor min
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Determination of error in interpolating polynomial
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Summing over continuum and uncountable numerocities
  • · Replies 1 ·
Replies
1
Views
3K
MHB Can the Series Sum Be Expressed as an Integral as N Approaches Infinity?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Express the limit as a definite integral
  • · Replies 16 ·
Replies
16
Views
4K
Graduate How to Derive a Closed Form for a Double Sum with Stochastic Variables?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Find ##\int_0^π \sin^ n x dx ##
  • · Replies 5 ·
Replies
5
Views
5K
Undergrad Finding the Largest (or smallest) Value of a Function - given some constant symmetric errors
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Limit of ##i^\frac{1}{n}## as ##n \to \infty##
  • · Replies 14 ·
Replies
14
Views
2K
Undergrad The Basic Area Problem (introduction to the topic of integrals)
  • · Replies 24 ·
Replies
24
Views
4K