Calculus Examples

$\sum_{k = 1}^{4} \frac{1}{k - k^{2}}$

Step 1

Simplify the summation.

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Step 1.1

Factor $k$ out of $k - k^{2}$ .

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Step 1.1.1

Raise $k$ to the power of $1$ .

$\frac{1}{k^{1} - k^{2}}$

Step 1.1.2

Factor $k$ out of $k^{1}$ .

$\frac{1}{k \cdot 1 - k^{2}}$

Step 1.1.3

Factor $k$ out of $- k^{2}$ .

$\frac{1}{k \cdot 1 + k (- k)}$

Step 1.1.4

Factor $k$ out of $k \cdot 1 + k (- k)$ .

$\frac{1}{k (1 - k)}$

Step 1.2

Rewrite the summation.

$\sum_{k = 1}^{4} \frac{1}{k (1 - k)}$

Step 2

Expand the series for each value of $k$ .

$\frac{1}{1 (1 - 1 \cdot 1)} + \frac{1}{2 (1 - 1 \cdot 2)} + \frac{1}{3 (1 - 1 \cdot 3)} + \frac{1}{4 (1 - 1 \cdot 4)}$

Step 3

Simplify.

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Step 3.1

Multiply $1 - 1 \cdot 1$ by $1$ .

$\frac{1}{1 - 1 \cdot 1} + \frac{1}{2 (1 - 1 \cdot 2)} + \frac{1}{3 (1 - 1 \cdot 3)} + \frac{1}{4 (1 - 1 \cdot 4)}$

Step 3.2

Multiply $- 1$ by $1$ .

$\frac{1}{1 - 1} + \frac{1}{2 (1 - 1 \cdot 2)} + \frac{1}{3 (1 - 1 \cdot 3)} + \frac{1}{4 (1 - 1 \cdot 4)}$

Step 3.3

Subtract $1$ from $1$ .

$\frac{1}{0} + \frac{1}{2 (1 - 1 \cdot 2)} + \frac{1}{3 (1 - 1 \cdot 3)} + \frac{1}{4 (1 - 1 \cdot 4)}$

Step 3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined