Infinite series involving 'x' has a constant value

  • #1
SatyaDas
22
0
How to prove that for all .
Using graph, we can see that the value of this series is 1 for all values of x.
[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-2.7634114187202106,"ymin":-4.543149781969371,"xmax":17.23658858127979,"ymax":8.437278166514556}},"randomSeed":"2ce535cd8c2a6f2ce21bf7669d621391","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"sum_{i=1}^{N}frac{1}{2^{3i}}left(csc^{2}left(frac{pi x}{2^{i}}right)+1right)sec^{2}left(frac{pi x}{2^{i}}right)sin^{2}left(pi xright)"},{"type":"text","id":"4","text":"N substitutes for infinity. Use slider to change value."},{"type":"expression","id":"3","color":"#388c46","latex":"N=9","hidden":true,"slider":{"hardMin":true,"hardMax":true,"min":"1","max":"20","step":"1"}},{"type":"expression","id":"2","color":"#2d70b3"}]}}[/DESMOS]
 
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  • #2
Use the result repeatedly to see that You can then write the th term of the series as Using that, you should be able to show by induction that the sum of the first terms of the series is That product of cosines lies between and . It follows that lies between and for all . Thus converges (uniformly) to the constant function .
 
  • #3
Great. Thanks.
How did you find the value of ?
 
  • #4
Satya said:
How did you find the value of ?
It's obvious from the graph at #1 that increases very rapidly to as increases. So it seemed helpful to write in the form - ?. I found that From there, it was easy enough to guess the general formula.
 

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