Let $ p_1<p_2 <cdots <p_k < cdots $ the increasing list in set $mathbb{P}$ of all prime numbers . By sum of infinite geometric series we have $sum
$$displaystyleprodlimits_ {i=1}^ {n} left (1+a_iright),, =,, displaystylesum_ {S,subseteq, {1,, 2,, 3,, dots,,, n}},,,left (,prod
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Absolute convergence - if $sum |a_n|$ converges, then $prod (1 + a_n)$ converges, and converges to the same value regardless of how the factors are rearranged.
The log of a product is the sum of logs of the things inside the product. So
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21 The symbol $Pi$ is the pi-product. It is like the summation symbol $sum$ but rather than addition its operation is multiplication. For example, $$ prod_
This integral can be solved easily (once you know the trick): you notice that ∏∞ n=0(1 +x2n) = 1 + x +x2 +x3 + ⋯ = 11−x ∏ n = 0 ∞ (1 + x 2 n) = 1 + x + 2 + 3 + = 1 1 x, and finally you have
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What does “$prod$” mean? Ask Question Asked 6 years, 11 months ago Modified 6 years, 11 months ago
Consider the functions f(x) = ∏∞ n=1(1 −xn) f (x) = ∏ n = 1 ∞ (1 x n) and g(x) = ∏∞ n=1(1 +xn) g (x) = ∏ n = 1 ∞ (1 + x n) f(x) f (x) is defined for x ∈ [−1, 1] x ∈ [1, 1] and g(x) g (x) is defined for
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I tried to prove the standard identities of the Dedekind eta function $$eta (tau)=q^ {frac {1} {24}} prod_ {n=1}^infty (1-q^n),$$ where $q=exp (2pi i tau)$ for some complex number $tau$,
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