Sequences and Series

The given series is related to the binomial expansion for any index. The binomial expansion for (1−x)−n when |x|<1 is given by :
(1−x)−n=1+nx+‌

n(n+1)

1⋅2

x2+‌

n(n+1)(n+2)

1⋅2⋅3

x3+...
To relate the series with the binomial expansion, consider setting x=‌

1

2

and n=‌

2

3

. The substitution into the binomial formula results in :
(‌

1

2

)−‌

2

3

=1+‌

2⋅1

3⋅2

+‌

2⋅5⋅1

3⋅6⋅4

+...
This series equates to ‌3√4 :
‌3√4=1+‌

2⋅1

3⋅2

+‌

2⋅5⋅1

3⋅6⋅4

+...
Thus, the assertion that this infinite series sums to ‌3√4 is correct, and it is derived from the application of the binomial series expansion with the appropriate substitutions for x and n.