Answer :
To determine the value of [tex]\( p \)[/tex] for the given series, let's analyze the terms of the series:
[tex]\[ 1 + \frac{4}{2^3} + \frac{9}{3^3} + \frac{16}{4^3} + \frac{25}{5^3} + \cdots \][/tex]
### Step-by-Step Simplification:
1. First, let's break down each term of the series:
- The first term is [tex]\( 1 \)[/tex].
- The second term is [tex]\( \frac{4}{2^3} \)[/tex]:
[tex]\[ \frac{4}{2^3} = \frac{4}{8} = \frac{1}{2} \][/tex]
- The third term is [tex]\( \frac{9}{3^3} \)[/tex]:
[tex]\[ \frac{9}{3^3} = \frac{9}{27} = \frac{1}{3} \][/tex]
- The fourth term is [tex]\( \frac{16}{4^3} \)[/tex]:
[tex]\[ \frac{16}{4^3} = \frac{16}{64} = \frac{1}{4} \][/tex]
- The fifth term is [tex]\( \frac{25}{5^3} \)[/tex]:
[tex]\[ \frac{25}{5^3} = \frac{25}{125} = \frac{1}{5} \][/tex]
2. Now, we see that each term follows the form [tex]\( \frac{i^2}{i^3} \)[/tex], which simplifies to [tex]\( \frac{1}{i} \)[/tex]. Therefore, we rewrite the initial few terms:
[tex]\[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots \][/tex]
3. Now let's assign the value [tex]\( p \)[/tex]. We observe that the series indeed simplifies as:
[tex]\[ \sum_{i=1}^{\infty} \frac{1}{i} \][/tex]
4. After simplifying the series, it is apparent that every term can be represented as [tex]\( \frac{1}{i} \)[/tex], meaning there is no multiplicative constant altering the series.
This leads us to the conclusion that the value of [tex]\( p \)[/tex] for this series is:
[tex]\[ p = 1 \][/tex]
Therefore, the value of [tex]\( p \)[/tex] is [tex]\( \boxed{1} \)[/tex].
[tex]\[ 1 + \frac{4}{2^3} + \frac{9}{3^3} + \frac{16}{4^3} + \frac{25}{5^3} + \cdots \][/tex]
### Step-by-Step Simplification:
1. First, let's break down each term of the series:
- The first term is [tex]\( 1 \)[/tex].
- The second term is [tex]\( \frac{4}{2^3} \)[/tex]:
[tex]\[ \frac{4}{2^3} = \frac{4}{8} = \frac{1}{2} \][/tex]
- The third term is [tex]\( \frac{9}{3^3} \)[/tex]:
[tex]\[ \frac{9}{3^3} = \frac{9}{27} = \frac{1}{3} \][/tex]
- The fourth term is [tex]\( \frac{16}{4^3} \)[/tex]:
[tex]\[ \frac{16}{4^3} = \frac{16}{64} = \frac{1}{4} \][/tex]
- The fifth term is [tex]\( \frac{25}{5^3} \)[/tex]:
[tex]\[ \frac{25}{5^3} = \frac{25}{125} = \frac{1}{5} \][/tex]
2. Now, we see that each term follows the form [tex]\( \frac{i^2}{i^3} \)[/tex], which simplifies to [tex]\( \frac{1}{i} \)[/tex]. Therefore, we rewrite the initial few terms:
[tex]\[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots \][/tex]
3. Now let's assign the value [tex]\( p \)[/tex]. We observe that the series indeed simplifies as:
[tex]\[ \sum_{i=1}^{\infty} \frac{1}{i} \][/tex]
4. After simplifying the series, it is apparent that every term can be represented as [tex]\( \frac{1}{i} \)[/tex], meaning there is no multiplicative constant altering the series.
This leads us to the conclusion that the value of [tex]\( p \)[/tex] for this series is:
[tex]\[ p = 1 \][/tex]
Therefore, the value of [tex]\( p \)[/tex] is [tex]\( \boxed{1} \)[/tex].
