Large Summation (Part 2)

Calculus Level 3

Which symbol should be used between the two summations? ∑ x = 1 ∞ 1 3 x ? ∑ x = 1 ∞ 1 4 x 2 − 1 \sum_{x=1}^\infty \frac{1}{3^{x}} \ \boxed{?} \ \sum_{x=1}^\infty \frac{1}{4x^{2}-1}

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Kaizen Cyrus by Kaizen Cyrus , 18, Philippines

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1 solution

∑ x = 1 ∞ 1 3 x = 1 3 1 − 1 3 = 1 2 \displaystyle \sum_{x=1}^\infty \dfrac {1}{3^x} = \dfrac {\frac 13}{1 - \frac 13} = \dfrac 12

∑ x = 1 ∞ 1 4 x 2 − 1 = 1 2 × ∑ x = 1 ∞ ( 1 2 x − 1 − 1 2 x + 1 ) = 1 2 × 1 ( 2 ( 1 ) − 1 ) = 1 2 \displaystyle \sum_{x=1}^\infty \dfrac {1}{4x^2-1} = \dfrac 12 \times \displaystyle \sum_{x=1}^\infty \begin{pmatrix} \dfrac {1}{2x-1} - \dfrac {1}{2x+1} \end{pmatrix} = \dfrac 12 \times \dfrac {1}{(2(1) - 1)} = \dfrac 12

Hence, ? = = \boxed{?} = \boxed {=}

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