Calculus Examples

$\sum_{k = 8}^{91} x^{k}$

Step 1

Split the summation to make the starting value of $k$ equal to $1$ .

$\sum_{k = 8}^{91} x^{k} = \sum_{k = 1}^{91} x^{k} - \sum_{k = 1}^{7} x^{k}$

Step 2

Evaluate $\sum_{k = 1}^{91} x^{k}$ .

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Step 2.1

Expand the series for each value of $k$ .

$x^{1} + x^{2} + x^{3} + \dots + x^{91}$

Step 2.2

Simplify the expanded form.

$0$

Step 3

Evaluate $\sum_{k = 1}^{7} x^{k}$ .

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Step 3.1

Expand the series for each value of $k$ .

$x^{1} + x^{2} + x^{3} + \dots + x^{7}$

Step 3.2

Simplify the expanded form.

$0$

Step 4

Replace the summations with the values found.

$0 - 0$

Step 5

Subtract $0$ from $0$ .

$0$