Calculus

Sequences and Series

Telescoping Series - Sum

         

Evaluate

1410+11016+11622+12228+12834.\frac{1}{4\cdot10}+\frac{1}{10\cdot16}+\frac{1}{16\cdot22}+\frac{1}{22\cdot28}+\frac{1}{28\cdot34} .

3123+5234+7345++81404142=? \frac{3}{1 \cdot 2 \cdot 3} + \frac{5}{2 \cdot 3 \cdot 4} + \frac{7}{3 \cdot 4 \cdot 5} + \cdots + \frac{81}{40 \cdot 41 \cdot 42} = ?

Evaluate

14+7+17+10++1397+400.\frac{1}{\sqrt{4}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{10}}+\cdots+\frac{1}{\sqrt{397}+\sqrt{400}}.

Suppose a,b,c,a,b,c, and dd are constants such that the following holds for all real numbers xx such that all denominators are nonzero: 10x(x+10)+10(x+5)(x+15)+10(x+10)(x+20)+10(x+15)(x+25)+10(x+20)(x+30)=a(x2+30x+75)x(x+b)(x+c)(x+d).\begin{aligned} & \frac{10}{x(x+10)}+\frac{10}{(x+5)(x+15)}+\frac{10}{(x+10)(x+20)} \\ &+ \frac{10}{(x+15)(x+25)}+\frac{10}{(x+20)(x+30)} \\ &= \frac{a(x^2+30x+75)}{x(x+b)(x+c)(x+d)}. \end{aligned} What is the value of a+b+c+d?a+b+c+d?

If f(x)=4x21,f(x)=4x^2-1, what is the value of

1f(1)+1f(2)+1f(3)++1f(84)?\frac{1}{f(1)}+\frac{1}{f(2)}+\frac{1}{f(3)}+\cdots+\frac{1}{f(84)}?

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