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?© BrainMass Inc. brainmass.com March 21, 2019, 10:55 pm ad1c9bdddf
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1. Note that finding the finite formula for the infinite sum
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
Dividing both sides by x, we ...
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.