Introduction
To explore the sum of the following series:
S(x) = Σ( f(N,x), N, 1, x)
As x varies from 1 to 250.
Goals
To explore different series to determine if the converge and at what approximate value. Each sum will be determined using 250 terms. I used the HP Prime to determine:
* The sum of each series after 250 terms: Σ(f(N,x), N, 1, 250), and,
* The change between the sum using 249 terms to the sum using 250 terms:
Σ(f(N,x), N, 1, 250) - Σ(f(N,x), N, 1, 249)
Decimal Accuracy for the HP Prime: 13 digits
* I also make a Scatterplot for each series using the Statistics 2Var app and turning the Fit option turned off. Plot setup: X Range = [-5, 255], Y Range = [-2, 2], X Tick = 10, Y Tick = 0.25
Each scatter plot is presented before the results.
Σ( 1/(1 - 2^N), N, 1, X)
250th Value: -1.60669515242
Change from 249th to 250th value: 0 (internal calculator accuracy has been met)
Σ( 1/(2^N - 1), N, 1, X)
250th Value: 1.60669515242
Change from 249th to 250th value: 0 (internal calculator accuracy has been met)
Σ( 1/(N^2), N, 1, X)
250th Value: 1.6409205624
Change from 249th to 250th value: 0.000016
If we allow this series to go to infinity, we get zeta(2) = π^2/6
Σ( 1/(2^N), N, 1, X)
250th Value: 1
Change from 249th to 250th value: 0 (internal calculator accuracy has been met)
Σ( 1/(2^N + N^2), N, 1, 250)
250th Value: 5.52714787527 x 10^-76
Change from 249th to 250th value: -5.527147875323 x 10^-76
Could this sum eventually converge to 0 as x approaches infinity?
This blog is property of Edward Shore. 2014
