Calculus Examples

$\sum_{1}^{50} \frac{1}{i}$

Step 1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 2

Substitute the values into the formula.

$(\frac{1}{i}) (50)$

Step 3

Simplify.

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Step 3.1

Multiply the numerator and denominator of $\frac{1}{i}$ by the conjugate of $i$ to make the denominator real.

$\frac{1}{i} \cdot \frac{i}{i} \cdot 50$

Step 3.2

Multiply.

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Step 3.2.1

Combine.

$\frac{1 i}{i i} \cdot 50$

Step 3.2.2

Multiply $i$ by $1$ .

$\frac{i}{i i} \cdot 50$

Step 3.2.3

Simplify the denominator.

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Step 3.2.3.1

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

$\frac{i}{i^{1} i} \cdot 50$

Step 3.2.3.2

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

$\frac{i}{i^{1} i^{1}} \cdot 50$

Step 3.2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{i}{i^{1 + 1}} \cdot 50$

Step 3.2.3.4

Add $1$ and $1$ .

$\frac{i}{i^{2}} \cdot 50$

Step 3.2.3.5

Rewrite $i^{2}$ as $- 1$ .

$\frac{i}{- 1} \cdot 50$

Step 3.3

Move the negative one from the denominator of $\frac{i}{- 1}$ .

$- 1 \cdot i \cdot 50$

Step 3.4

Rewrite $- 1 \cdot i$ as $- i$ .

$- i \cdot 50$

Step 3.5

Multiply $50$ by $- 1$ .

$- 50 i$