Consider the following function: $f(x) = \frac{7 \cos(\pi x)}{\sqrt{x}}.$

What conclusions can be made about the series: $\displaystyle\sum_{n=1}^\infty \frac{7 \cos(\pi n)}{\sqrt{n}}$ and the Integral Test?

Neetesh Kumar | December 27, 2024

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This is the solution to Math 1c
Assignment: 11.3 Question Number 10
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Step-by-step solution:

Step 1: Analyze the given function:

The given function is:

$f(x) = \frac{7 \cos(\pi x)}{\sqrt{x}}.$

Properties of $f(x)$:

1. Oscillating Behavior: The $\cos(\pi x)$ term oscillates between $-1$ and $1$. This means $f(x)$ is not always positive or monotonic on the interval $[1, \infty)$.
2. Continuity: The function is continuous for $x > 0$, but the oscillating term prevents the function from being decreasing over $[1, \infty)$.

Step 2: Applicability of the Integral Test:

The Integral Test requires the function $f(x)$ to be:

• Positive,
• Continuous,
• Decreasing on $[1, \infty)$.

Since $f(x)$ is not positive and not monotonic (because of the oscillating $\cos(\pi x)$ term), the Integral Test cannot be applied to determine the convergence or divergence of the series:

$\displaystyle\sum_{n=1}^\infty \frac{7 \cos(\pi n)}{\sqrt{n}}.$

Final Answer:

$\boxed{\text{The Integral Test cannot be used to determine whether the series converges, as the function} \space f(x) \space \text{is not positive and not decreasing on} \space [1, \infty)}$.

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