Q. 92 - Calculus

Question

 Prove that every sequence of the form ${{\left\{a_k\right\}}_{k=n}}^{\infty }$can be rewritten as a sequence of the form ${{\left\{a_k\right\}}_{k=1}}^{\infty }$.

Step-by-Step Solution

Answer

Proved

Step 1. Given

Consider the sequence ${{\left\{a_k\right\}}_{k=n}}^{\infty }$.

Step 2. Proof

$$\begin{gathered} \mathrm{As} \mathrm{the} \mathrm{given} \mathrm{sequence} {{\left\{{\mathrm{a}}_{\mathrm{k}}\right\}}_{\mathrm{k}=\mathrm{n}}}^{\infty } \mathrm{starts} \mathrm{from} \mathrm{n} \mathrm{to} \infty \\ \mathrm{Since} \mathrm{the} \mathrm{sequence} \mathrm{is} \mathrm{tending} \mathrm{to} \mathrm{infinity} \\ \mathrm{so} \mathrm{if} \mathrm{we} \mathrm{start} \mathrm{k}=1 \mathrm{then} \mathrm{the} \mathrm{sequence} \mathrm{makes} \mathrm{no} \mathrm{difference} \\ Hence proved \\ {{\left\{a_k\right\}}_{k=n}}^{\infty }={{\left\{a_k\right\}}_{k=1}}^{\infty } \end{gathered}$$