Answer to Question #184021 in Real Analysis for Tulela

Question

If a is a sequence of real numbers, then

△a = (an+1 − an)N

is called the difference sequence of a.

a) Let a be a sequence of real numbers. Find △2a := △(△a)

 b) If a is a convergent sequence of real numbers, prove that △a is a null se- quence.

Answer 1

$△a_n = (a_{n+1} − a_n)$

(a) $△^2a=\Delta a_{n+1}-\Delta a_n$

$=(a_{n+2}-a_{n+1})-(a_{n+1}-a_n)\\=a_{n+2}-2a_{n+1}+a_n$

(b) a is a convergent sequence of real numbers

Then the value of a is finite, Hence $\Delta a$ is null sequence.