Calculus Examples

$\sum_{n = 1}^{\infty} \frac{1}{n}$

Step 1

To determine if the series is convergent, determine if the integral of the sequence is convergent.

$\int_{1}^{\infty} \frac{1}{x} d x$

Step 2

Write the integral as a limit as $t$ approaches $\infty$ .

$lim_{t \to \infty} \int_{1}^{t} \frac{1}{x} d x$

Step 3

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$lim_{t \to \infty} \ln (| x |)]_{1}^{t}$

Step 4

Simplify the answer.

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Step 4.1

Evaluate $\ln (| x |)$ at $t$ and at $1$ .

$lim_{t \to \infty} (\ln (| t |)) - \ln (| 1 |)$

Step 4.2

Remove parentheses.

$lim_{t \to \infty} \ln (| t |) - \ln (| 1 |)$

Step 4.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$lim_{t \to \infty} \ln (\frac{| t |}{| 1 |})$

Step 5

As log approaches infinity, the value goes to $\infty$ .

$\infty$

Step 6

Since the integral is divergent, the series is divergent.

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