Calculus Examples

$\sum_{i = 1}^{10} 5 + \frac{1}{2 i}$

Step 1

Split the summation into smaller summations that fit the summation rules.

$\sum_{k = 1}^{10} 5 + \frac{1}{2 i} = \sum_{k = 1}^{10} 5 + \sum_{k = 1}^{10} \frac{1}{2 i}$

Step 2

Evaluate $\sum_{k = 1}^{10} 5$ .

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Step 2.1

The formula for the summation of a constant is:

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

Step 2.2

Substitute the values into the formula.

$(5) (10)$

Step 2.3

Multiply $5$ by $10$ .

$50$

Step 3

Evaluate $\sum_{k = 1}^{10} \frac{1}{2 i}$ .

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Step 3.1

Expand the series for each value of $i$ .

$\frac{1}{2 \cdot 1} + \frac{1}{2 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{2 \cdot 10}$

Step 3.2

Simplify the expanded form.

$\frac{7381}{5040}$

Step 4

Add the results of the summations.

$50 + \frac{7381}{5040}$

Step 5

Simplify.

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Step 5.1

To write $50$ as a fraction with a common denominator, multiply by $\frac{5040}{5040}$ .

$50 \cdot \frac{5040}{5040} + \frac{7381}{5040}$

Step 5.2

Combine $50$ and $\frac{5040}{5040}$ .

$\frac{50 \cdot 5040}{5040} + \frac{7381}{5040}$

Step 5.3

Combine the numerators over the common denominator.

$\frac{50 \cdot 5040 + 7381}{5040}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $50$ by $5040$ .

$\frac{252000 + 7381}{5040}$

Step 5.4.2

Add $252000$ and $7381$ .

$\frac{259381}{5040}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{259381}{5040}$

Decimal Form:

$51.46448412 \dots$

Mixed Number Form:

$51 \frac{2341}{5040}$