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finding the finite formula for the infinite sum

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1) When is sin(x)=0? Why is it plausible that sin(x)=x(1-x/pi)(1+x/pi)(1-x/2*pi)(1+x/2*pi) ... ?

2) Explain how Euler concludes from 1) that pi^2/6=1+1/4+1/9+1/16 ... .

3) Compare the formula in 2) to Archimedes' calculation for pi. Which of these methods is more efficient and why?

4) Why does the argument from 3) not work to show the formula in 1) is correct?

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Solution Summary

The finite formula for the infinite sum is demonstrated, as other the steps for solving various other basic algebra problems related to the work of Euler and Archimedes.

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1. Note that finding the finite formula for the infinite sum
1+1/4+1/9+1/16+...
is known as the Basel problem. Euler finally showed in 1741 that this sum converges to (pi^2)/6.

2) Recall that the Maclaurin series for sin??(x)? is given by
sin?(x)=x-x^3/3!+x^5/5!-x^7/7!+?
Dividing both sides by x, we ...

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