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

Hi,

ReplyDeletewhat is your absolute favorite (if sum everything) - HP Prime or Casio Prizm? :)

Vitalisam,

ReplyDeleteGood and very though question. I'll go with the Prime because I am using it the most right now and currently it is my go-to calculator.

Eddie

Thanks!

DeleteI tend to agree with you about Prime.

But I have a dream - tiny, light and fast programmable calculator with high resolution black/white eInk display, powered from two AAA batteries for few months :)