b) III only
S2 and S3 are divergent
Question 2:
Side length:
Total area:
Total area:
Side length:
Total area:
Solution: Let’s try your favourite values
and
For
we have
Converges.
Suppose that
and
where c is a finite positive
If
converges and c = 0, then
If
converges as well.
diverges and
If
then
diverges as well.
Solution: For series III
Consider
Then
Since
diverges, series III diverges as well.
Since the series
converges as well.
For series II
Consider
Then
the series
converges,
Since the geometric series
converges as well.
For series I
Consider
Then
the series
converges
For example, for the series and
the series of absolute values are
series by taking absolute values:
given by and
we say that the series converges absolutely.
series converges, then the series
converges as well.
then we say that the series
converges conditionally.
diverges,
If
where
Solution: For series I:
and the sequence an does not converge to 0.
Thus, the necessary condition of convergence is not satisfied. Hence, series I diverges.
For series III:
Let’s compare |an| with
Hence, series III converges absolutely.
and
The series
converges.
Consider the function
correct
is non-increasing, for n = 1,2,3…
Solutions:
for
Hence, the sequence
decreases for n = 1,2,3…
Thus, the series
converges conditionally.
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