Suppose is a sequence of functions satisfying for in an interval and _ k=0 ^ (-1)^k…

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Suppose

\{a_k(x)\}_{k\ge 0}

is a sequence of functions satisfying

a_k(x)>0

for

x

in an interval

I

and

\sum_{k=0}^{\infty} (-1)^k a_k(x)

converges for every

x\in I

. Assume further that

a_{k+1}(x)\le a_k(x)

for all

k\ge 0

and all

x\in I

, and

\displaystyle \lim_{k\to\infty} a_k(x)=0

. Let

S_n(x)

denote the partial sum through

k=n

and

R_n(x)=\bigl|\,S(x)-S_n(x)\,\bigr|

the remainder at order

n

. (a) Explain, without performing any explicit calculations, why the single inequality

a_{n+1}(x)<\varepsilon

ensures that

R_n(x)<\varepsilon

for all

x\in I

. Your explanation should rely only on the stated properties of the sequence

\{a_k(x)\}

. (b) Discuss qualitatively how the size of

x

might influence the minimal integer

n

required to achieve a given tolerance

\varepsilon

. Your answer should reference the behaviour of the functions

a_k(x)

but need not involve explicit formulas.