Answer :
To determine the value of [tex]\( p \)[/tex] for the given [tex]\( p \)[/tex]-series, we start by analyzing the series:
[tex]\[ \sum_{n=1}^{\infty} \frac{3}{n^2 \cdot n^{1/3}} \][/tex]
First, we simplify the denominator. The expression [tex]\( n^2 \cdot n^{1/3} \)[/tex] can be rewritten using the properties of exponents:
[tex]\[ n^2 \cdot n^{1/3} = n^{2 + 1/3} \][/tex]
This simplification gives us:
[tex]\[ n^{2 + 1/3} = n^{\frac{6}{3} + \frac{1}{3}} = n^{\frac{7}{3}} \][/tex]
Now, we substitute this back into the series:
[tex]\[ \sum_{n=1}^{\infty} \frac{3}{n^{7/3}} \][/tex]
Next, we recall the general form of a [tex]\( p \)[/tex]-series:
[tex]\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \][/tex]
By comparing the simplified series with the general form, we observe that [tex]\( p \)[/tex] in our case is the exponent in the denominator's [tex]\( n \)[/tex]-term. For the given series:
[tex]\[ \sum_{n=1}^{\infty} \frac{3}{n^{7/3}} \][/tex]
we can see that [tex]\( p = \frac{7}{3} \)[/tex].
Converting the fraction [tex]\(\frac{7}{3} \)[/tex] to a decimal, we get:
[tex]\[ \frac{7}{3} \approx 2.3333333333333335 \][/tex]
Hence, the value of [tex]\( p \)[/tex] for the given [tex]\( p \)[/tex]-series is:
[tex]\[ \boxed{2.3333333333333335} \][/tex]
[tex]\[ \sum_{n=1}^{\infty} \frac{3}{n^2 \cdot n^{1/3}} \][/tex]
First, we simplify the denominator. The expression [tex]\( n^2 \cdot n^{1/3} \)[/tex] can be rewritten using the properties of exponents:
[tex]\[ n^2 \cdot n^{1/3} = n^{2 + 1/3} \][/tex]
This simplification gives us:
[tex]\[ n^{2 + 1/3} = n^{\frac{6}{3} + \frac{1}{3}} = n^{\frac{7}{3}} \][/tex]
Now, we substitute this back into the series:
[tex]\[ \sum_{n=1}^{\infty} \frac{3}{n^{7/3}} \][/tex]
Next, we recall the general form of a [tex]\( p \)[/tex]-series:
[tex]\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \][/tex]
By comparing the simplified series with the general form, we observe that [tex]\( p \)[/tex] in our case is the exponent in the denominator's [tex]\( n \)[/tex]-term. For the given series:
[tex]\[ \sum_{n=1}^{\infty} \frac{3}{n^{7/3}} \][/tex]
we can see that [tex]\( p = \frac{7}{3} \)[/tex].
Converting the fraction [tex]\(\frac{7}{3} \)[/tex] to a decimal, we get:
[tex]\[ \frac{7}{3} \approx 2.3333333333333335 \][/tex]
Hence, the value of [tex]\( p \)[/tex] for the given [tex]\( p \)[/tex]-series is:
[tex]\[ \boxed{2.3333333333333335} \][/tex]
