Asked by Ian
If {an} (a sequence) is decreasing and an > 0 for all n, then {an} is convergent. True/False?
Answers
Answered by
Steve
nope. consider {an} = 1/n
The harmonic series diverges.
The harmonic series diverges.
Answered by
David
True.
When considering a SEQUENCE, such as {an} = 1/n^2, it will always be greater than 0, yet by an infinitely small amount as n approaches infinity. This is why we say it converges at 0, although it never actually reaches it.
{an} = 1/n the SEQUENCE will behave the same way. However, it is important to distinguish between series and sequences. The harmonic SERIES 1/n is the sum of every nth term, and will approach infinity.
When considering a SEQUENCE, such as {an} = 1/n^2, it will always be greater than 0, yet by an infinitely small amount as n approaches infinity. This is why we say it converges at 0, although it never actually reaches it.
{an} = 1/n the SEQUENCE will behave the same way. However, it is important to distinguish between series and sequences. The harmonic SERIES 1/n is the sum of every nth term, and will approach infinity.
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