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∑n=1∞1n2+3n+2= ?\large \displaystyle \sum_{n = 1}^{\infty} \dfrac {1}{n^2 + 3n + 2} = \ ? n=1∑∞n2+3n+21= ?
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1+12+16+112+120+…= ?1 + \frac {1}{\color{#3D99F6}2} + \frac {1}{\color{#3D99F6}6} + \frac {1}{\color{#3D99F6}{12}} + \frac {1}{\color{#3D99F6}{20}} + \ldots = \ \color{teal}? 1+21+61+121+201+…= ?
Consider the following pattern:
11×2=11−1212×3=12−1313×4=13−14⋮\begin{aligned} \frac{1}{1\times 2} & = \frac{1}{1} - \frac{1}{2} \\ & \\ \frac{1}{2\times 3} & = \frac{1}{2} - \frac{1}{3} \\ & \\ \frac{1}{3\times 4} & = \frac{1}{3} - \frac{1}{4} \\ \vdots & \end{aligned} 1×212×313×41⋮=11−21=21−31=31−41
Following the pattern above, if 111×12=1a−1b, \displaystyle \frac{1}{11\times 12} = \frac{1}{a} - \frac{1}{b}, 11×121=a1−b1, what are the values of a a a and b bb?
11×2+12×3+13×4+⋯+199×100= ? \large \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\cdots +\frac{1}{99 \times 100} = \ ? 1×21+2×31+3×41+⋯+99×1001= ?
11⋅3+13⋅5+15⋅7+17⋅9+⋯= ?\large \frac{1}{1\cdot 3} + \frac{1}{3\cdot 5} +\frac{1}{5\cdot 7} + \frac{1}{7 \cdot 9} + \cdots = \ ?1⋅31+3⋅51+5⋅71+7⋅91+⋯= ?
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Consider the following pattern: 
