Question

1) if a sequence is monotone decreasing and greater than 0 for all values of n (n=1 to infinity) then the sequence must converge. True or false?

2) In order for infinite series k=1 to infinity (ak + bk) = series ak + series bk, both series must converge. True or false?

3) Let f(x) be a continuous decreasing function where f(k) = ak. If integral 1 to infinity f(x) = 5, what can we conclude about series ak?

-series must converge

-must diverge

- must converge to 5

- no conclusion can be made

Answer #1

1. The Taylor series for f(x)=x^3 at 1 is ∞∑n=0 cn(x−1)^n.
Find the first few coefficients.
c0=
c1=
c2=
c3=
c4=
2. Given the series:
∞∑k=0 (−1/6)^k
does this series converge or diverge?
diverges
converges
If the series converges, find the sum of the series:
∞∑k=0 (−1/6)^k=

1. Test the series below for convergence using the Root
Test.
∞∑n=1 (4n/10n+1)^n
The limit of the root test simplifies to lim n→∞ |f(n)| where
f(n)=
The limit is:
Based on this, the series
Diverges
Converges
2. We want to use the Alternating Series Test to determine if
the series:
∞∑k=4 (−1)^k+2 k^2/√k^5+3
converges or diverges.
We can conclude that:
The Alternating Series Test does not apply because the terms of
the series do not alternate.
The Alternating Series Test...

3. Consider the following property: for any ε>0, there exists
N∈N so that whenever n≥N,|u_n+1−u_n|<ε.
What is the difference between this property and the definition
of a Cauchy sequence?
Find a convergent sequence which has this property.
Find a divergent sequence which has this property. (Hint: can
you think of a function f(x) which grows to infinity very
slowly? Then try a_n=f(n).

Consider the series ∑n=1 ∞ an
where
an=(5n+5)^(9n+1)/
12^n
In this problem you must attempt to use the Ratio Test to decide
whether the series converges.
Compute
L= lim n→∞
∣∣∣an+1/an∣∣
Enter the numerical value of the limit L if it converges, INF if
the limit for L diverges to infinity, MINF if it diverges to
negative infinity, or DIV if it diverges but not to infinity or
negative infinity.
L=
Which of the following statements is true?
A. The...

We are given a sequence of numbers: 1, 3, 5, 7, 9, . . . and
want to prove that the closed formula for
the sequence is an = 2n – 1.
What would the next number in the sequence be?
What is the recursive formula for the
sequence?
Is the closed formula true for
a1?
What about a2?
What about a3?
Critical Thinking
How many values would we have to check before we could be sure
that the...

Select "True" or "False" for the sentence or statement.
1. All relative extrema occur at first - order cirtical
values
2. Some points of inflection do not occur ar second - order
critical values.
3. If the slopes of the tangent lines of f(x) are positive on
(a,b) then f(x) is decreasing on (a,b)
4. If f(x) is continuous on the closed interval (a,b) then f(x)
must have an absolute minimum and an absolute maximum on the
interval

1. (10 pts) If A and B are independent events (both with
probability greater than 0) then which of the following statements
must be true?
A. P(A) + P(B) = 1
B. P(A and B) = P(A)P(B)
C. P(A and B) = 0
D. P(A and B) = P(A) + P(B)
2. (10 pts)A salesperson makes “cold calls” trying to sell a
product by phone and is successful on each call with probability
1/50. Whether or not he is successful...

1.. Protein X has a single N-terminal ER signal sequence, while
protein Y has a single internal ER signal sequence. Which of these
two proteins will be found in the ER lumen (Answer "X" or "Y")?
2. Protein A has an N-terminal ER signal sequence and one stop
transfer sequence, while Protein B has an internal ER signal
sequence and one stop transfer sequence. Which of these two
proteins will have two transmembrane domains once its translocation
is complete (Answer...

1.. Protein X has a single N-terminal ER signal sequence, while
protein Y has a single internal ER signal sequence. Which of these
two proteins will be found in the ER lumen (Answer "X" or "Y")?
2. Protein A has an N-terminal ER signal sequence and one stop
transfer sequence, while Protein B has an internal ER signal
sequence and one stop transfer sequence. Which of these two
proteins will have two transmembrane domains once its translocation
is complete (Answer...

True or False:
10. The probability of an event is a value which must be greater
than 0 and less than 1.
11. If events A and B are mutually exclusive, then P(A|B) is always
equal to zero.
12. Mutually exclusive events cannot be independent.
13. A classical probability measure is a probability assessment
that is based on relative frequency.
14. The probability of an event is the product of the probabilities
of the sample space outcomes that correspond to...

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