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# It is known that sin πχ n2 The expression on the right is an infinite product, which needs to be ...

## Question

###### It is known that sin πχ n2 The expression on the right is an infinite product, which needs to be ... It is known that sin πχ n2 The expression on the right is an infinite product, which needs to be interpreted as the limit of the corresponding finite products. Notice that for a given value of x, the value of the nth factor approaches 1 as n approaches infinity Notice also that the infinite product vanishes whenever x is an integer; when x = ±n, the nth factor is zero The expression on the left also vanishes whenever x is an integer, since the sine of any whole multiple of π is zerO Both sides of the equation (1) have power series expansions. On the left, Taylor's formula gives sin πχ EO (2i+1)! The one on the right is more complicated. It is a power series in x in which the the coefficients themselves are infinite series. The first two terms are You will need to find the next two terms (meaning the coefficients of x and x') yourself When two power series are equal, each coefficient of one of them is equal to the corresponding coefficient of the other one. Comparing the coefficients of x3 gives us the formula n2 6 n-1 which was originally proved by Leonhard Euler in 1735 1. ((1/(N+1)) points) Use similar methods to show c0 n 90 945 n= 1 に!

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