Determine the convergence or divergence of the p-series. 1 + 1/4 Squareroot 2 +1/9 Squareroot...

Question:

Determine the convergence or divergence of the p-series.

{eq}1+\frac{1}{4\sqrt{2}}+\frac{1}{9\sqrt{3}}+\frac{1}{16\sqrt{4}}+\frac{1}{25\sqrt{5}}+... {/eq}

P-Series:

If {eq}{a_1},{a_2},{a_3},... {/eq} is a sequence of real number then the infinite sum

{eq}{a_1} + {a_2} + {a_3} + ... {/eq} is called a series of the real number.

Generally, series is denoted by, {eq}\sum {{a_n}} {/eq}

P-Test:

If the series is in the form {eq}\sum {\frac{1}{{{n^p}}}} {/eq}

is called p-series. {eq}(p>0) {/eq}

Consider series {eq}\sum\limits_{n = 1}^\infty {\frac{1}{{{n^p}}}} = 1 + \frac{1}{{{n^2}}} + \frac{1}{{{n^3}}} + ... {/eq} where {eq}p > 0 {/eq}

Then we have the following discussion,

{eq}\eqalign{ & 1)\text{if}~P > 1~\text{then series is convergent} \cr & 2)\text{if}~p \leqslant 1~\text{then series is divergent} \cr} {/eq}

Answer and Explanation: 1

Consider the given series,

{eq}1+\frac{1}{4\sqrt{2}}+\frac{1}{9\sqrt{3}}+\frac{1}{16\sqrt{4}}+\frac{1}{25\sqrt{5}}+.... {/eq}

The above series can be written as,

{eq}\eqalign{ 1 + \frac{1}{{4\sqrt 2 }} + \frac{1}{{9\sqrt 3 }} + \frac{1}{{16\sqrt 4 }} + \frac{1}{{25\sqrt 5 }} + .... &= 1 + \frac{1}{{{2^2}{2^{\frac{1}{2}}}}} + \frac{1}{{{3^2}{3^{^{\frac{1}{2}}}}}} + \frac{1}{{{4^2}{4^{^{\frac{1}{2}}}}}} + \frac{1}{{{5^2}{5^{^{\frac{1}{2}}}}}} + .... \cr & = \frac{1}{{{1^0}}} + \frac{1}{{{2^{2 + \frac{1}{2}}}}} + \frac{1}{{{3^{2 + \frac{1}{2}}}}} + \frac{1}{{{4^{2 + \frac{1}{2}}}}} + \frac{1}{{{5^{2 + \frac{1}{2}}}}} + ... \cr & = \frac{1}{{{1^{\frac{5}{2}}}}} + \frac{1}{{{2^{\frac{5}{2}}}}} + \frac{1}{{{3^{\frac{5}{2}}}}} + \frac{1}{{{4^{^{\frac{5}{2}}}}}} + \frac{1}{{{5^{^{\frac{5}{2}}}}}} + ... \cr & = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^{\frac{5}{2}}}}}} \cr} {/eq}

Here {eq}p= \frac{5}{2} > 1 {/eq}hence by p-test given series is convergent.


Learn more about this topic:

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P-Series: Definition & Examples

from

Chapter 29 / Lesson 5
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This lesson is designed to help you understand a specific type of series called a p-series. You will determine if a series is a p-series, and you will learn to decide if a p-series converges or diverges.


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