Math 166Z Homework # 7
Due Thursday March 24th
1. Consider the sequence an =
n+1
n
which converges to 1.
(a) find a number N such that |an − 1| < .1 for all n ≥ N .
(b) find a number N such that |an − 1| < .01 for all n ≥ N .
(c) find a number N such that |an − 1| < .001 for all n ≥ N .
2. Does the sequence an = ln(n + 1) − ln(n) converge? If so, what number does an converge
to?
1 + 2 + 3 + ··· + n
converges to 12 .
n2
Hint: Can you think of another way to write 1 + 2 + 3 + · · · + n? (you might want to
look in section 4.1)
3. Show that the sequence an =
4. For the following, determine whether the series is convergent of divergent. If it is convergent, find its sum.
∞
X
5n
(a)
8n
n=0
(b)
(c)
∞
X
4n+1
n=0
∞
X
n=1
(d)
5n
1
2n
∞
X
2n
n=3
7n
5. For the following, find the values of x for which the series converges. What is the sum
of the series for these values of x?
∞
X
(a)
(x − 3)n
(b)
n=0
∞
X
n=2
xn
5n
6. Write the following repeating decimals in the form of ab where a and b are integers.
Hint: Re-write them as an infinite series and then find the sum of that series.
(a) 0.1 = 0.11111 . . .
(b) 0.123 = 0.1232323 . . .
(c) 0.045 = 0.0454545 . . .
(d) 0.19 = 0.199999 . . .
7. The present value (PV) of a payment in the future is given by the formula
x
PV =
(1 + i)n
where:
x = the amount of the payment amount,
i = the interest rate
n = the number of time periods in the future the payment is made
If I decide to give you $100 every year from now on (forever, these payments will
eventually be made to your decedents) and the interest rate is 5% per year, what is
the present value of these payments?
PV =
∞
X
n=1
100
(1 + .05)n
(It may seem kind of strange, but this kind of payment was used in the past. It is called
a perpetuity. The government of the United Kingdom used them in the 1750s to help
reduce what it had to pay on its debt. It is still paying some of them today.)
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