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00 13. Suppse that an is a convergent series with known sum L. Let S = ax be then the partiul sum for this series. a) (a) Find lim S. +00 (b) Find limo 0. (e) Find lim S. d) Find lim 100T 0

Answer :

Partial sums are:

a) limx→∞ S = L
b) The limit does not exist.
c) limx→∞ S = L
d) The limit does not exist.

How did this partial sums evaluate?

We need to use the formulas for partial sums and limits of sequences.

First, recall that the nth partial sum of a series is given by:

Sn = a1 + a2 + ... + an

And the limit of a sequence (if it exists) is given by:

limn→∞ an

Now, let's use these formulas to answer the parts of the question:

a) Find lim S as n approaches infinity:

We have:

S = ax = a1 + a2 + a3 + ... + ax

Taking the limit as x approaches infinity, we get:

limx→∞ S = limx→∞ (a1 + a2 + a3 + ... + ax) = limn→∞ Sn

But we know that the series is convergent, so the limit of the partial sums exists and is equal to the sum of the series:

limn→∞ Sn = L

Therefore:

limx→∞ S = L

b) Find lim as x approaches 0:

We have:

S = ax = a1 + a2 + a3 + ... + ax

Taking the limit as x approaches 0, we get:

limx→0 S = limx→0 (a1 + a2 + a3 + ... + ax)

But as x approaches 0, the number of terms in the sum approaches infinity, so this limit does not exist.

c) Find lim S as x approaches infinity:

We have:

S = ax = a1 + a2 + a3 + ... + ax

Taking the limit as x approaches infinity, we get:

limx→∞ S = limx→∞ (a1 + a2 + a3 + ... + ax) = limn→∞ Sn

Again, we know that the limit of the partial sums exists and is equal to the sum of the series:

limn→∞ Sn = L

Therefore:

limx→∞ S = L

d) Find lim as x approaches 100:

We have:

S = ax = a1 + a2 + a3 + ... + ax

Taking the limit as x approaches 100, we get:

limx→100 S = limx→100 (a1 + a2 + a3 + ... + ax)

But as x approaches 100, the number of terms in the sum approaches infinity, so this limit does not exist.


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