Mathematics

Problem

Which of the following sets of conditions must be verified in order to guarantee that the remainder

R_N

of the alternating series

\displaystyle\sum_{n=1}^{\infty} (-1)^{\,n+1}a_n

satisfies

|R_N|\le a_{N+1}

? a)

a_n>0

for all

n

and

\displaystyle\lim_{n\to\infty}a_n=0

. b)

a_n>0

for all

n

\{a_n\}

is increasing, and

\displaystyle\lim_{n\to\infty}a_n=0

. c)

a_n>0

for all

n

\{a_n\}

is decreasing, and

\displaystyle\lim_{n\to\infty}a_n=0

. d) The terms alternate in sign and

\sum a_n

is convergent by any test.