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Question 1
Write the first three terms of .
= 64, 182.25, 400, …
Evaluate the limit of the following sequence
The sequence can be viewed as sum of two convergent sequences, namely and . Thus,

Note that , therefore
Both the limit of the left and the right side of the inequality are
Therefore, = 0
Note that
The terms are all less than or equal to 1:
Therefore,
Then, the inequality is

According to the squeezing theorem, set
Continue…
Both the limit of the left and the right side of the inequality are
Therefore,

Determine whether the following series is convergent or divergent by applying relevant tests

The general term for the series is
Consider the integral of the function over the interval [1,
The integral is convergent. From the Cauchy’s integral test, the series is convergent.
The limit of the root of the term is

According to the alternating series test, it satisfied both of the following conditions:

Question 2
Evaluate the sum of the following series:

1 + 2 – 3 + 4 – 5 + 6 – 7 + … + 2000
Arrange series into odd and even

[even]
Write the sum twice, one in an ordinary order and the other in a reverse order: By adding vertically, each pair of numbers adds up to 2002
Find n to find out how many number of terms in the series
Since there are 1000 of these sums of 2002

[odd]
By adding vertically, each pair of numbers adds up to 2002
Since there are 999 of these sums of 2002

(i) Show that converges absolutely.
Note that each term in this satisfies

p = 2 >1. Therefore, from the comparison test, can conclude that
is convergent which implies that converges
absolutely.
Show that the series is conditionally convergent.
Note that this is an alternating series. The terms in the series satisfy the conditions in the Leibniz’s theorem:

Continue…
But the series is not absolutely convergent : the series

The term on the left side of the inequality is…

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