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whitejac
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Homework Statement
"For the given series, write formulas for the sequences an , Sn, Rn and find the limit as n->∞ (if it exists)
Homework Equations
∑∞1 ((1/n) - 1/(n+1)
The Attempt at a Solution
I know how to take the limit, that's no problem. I'm a bit confused about what an , Sn, Rn are referring to. Maybe I'm just misunderstanding the usage...
My book speaks generally of defining them (by that i mean, no formal definition is given but it often refers to a previous example) however it does say this:
Let us call the terms of the series a so that the series
is
"a1 + a2 + a3 + a4 + · · · + ann + · · · ."
Remember that the three dots mean that there is never a last term; the series goes
on without end. Now consider the sums Sn that we obtain by adding more and
more terms of the series. We define
S1 = a1,
S2 = a1 + a2,
S3 = a1 + a2 + a3,
· · ·
Sn = a1 + a2 + a3 + · · · + an
It then goes further to say:
The difference Rn = S − Sn is called the remainder (or the remainder after n
terms). From (4.6), we see that...
Where S is the lim n-> ∞. So can Rn not be found if there's no limit?
Also, why even write down an if it's just the same as Sn?