Determine if the sequence an*= {3/4, 1/8, -1/2, -9/8...} converges or diverges
then determine if the associated series 3/4 +1/8+ (-1/2)+(-9/8)... converges or diverges
*the n is in subscript
1. D) The sequence diverges; the series diverges
2. D) geometric, divergent
3. C & E) a geometric sequence with r = (3/5) ; a geometric sequence with r = -(1/6)
might be different for non-honors
honors student is correct
hard to say. There is no common ratio, and I see no useful pattern in the fractions.
The denominators are all powers of 2, but they are 2,3,1,3 ...
If we use a common denominator of 8, the numerators are 6,1,4,9 ...
I guess you could separate it into two geometric sequences, with
a = 3/4 and r = 1/6
and
a = -1/2 and r = 9/4
Those would both converge, and the sums are easy to calculate.
Other than that, I got nothing.
Going with ooblecks idea of finding a common denominator of 8, we get
{3/4, 1/8, -1/2, -9/8...}
= {6/8, 1/8, -4/8, -9/8...} , which is an arithmetic sequence, with common difference of -5/8
What do you know about convergence and divergence of arithmetic sequences ?
Well, this sequence seems to be confusing. You know what they say - sequences that make you go "huh?" are the most entertaining.
For clarity, let's rewrite the sequence:
an* = {3/4, 1/8, -1/2, -9/8...}
Now, to determine if the sequence converges or diverges, we need to check if it approaches a specific value or if it keeps going off into infinity. It's like trying to find a runaway clown at the circus. Will they catch it in time? Or will it continue to juggle its way out of control?
Looking at the sequence, it seems to be getting smaller and smaller. In fact, as we move along, each term becomes smaller in magnitude. It's like watching a shrinking clown balancing on a tiny unicycle. So, it appears like the sequence might be converging.
But let's double-check our clown-clusion by taking the limit as n approaches infinity:
lim (n→∞) an* = lim (n→∞) (-(n+1)/2^n)
= lim (n→∞) -(n+1)/2^n
Now, let's send in a team of circus mathematicians to investigate this limit. After careful calculations, they found that the limit is equal to... **drumroll please**... 0!
So, the sequence an* converges to 0, like a runaway clown finally coming to a halt. But our circus show doesn't end there!
Now, we need to determine if the associated series converges. Fancy that!
The series 3/4 + 1/8 + (-1/2) + (-9/8) + ... is just the sum of the terms in the sequence. And since we clown-cluded that the sequence converged to 0, it's like saying we have an unending circus where all the clowns eventually sum up to zero.
Therefore, the associated series 3/4 + 1/8 + (-1/2) + (-9/8) + ... converges to 0! Hooray for circus math and clown convergence!
I hope that brought a smile to your face, just like seeing a clown stumble gracefully on their oversized shoes.
To determine if a sequence converges or diverges, we need to analyze the behavior of its terms as n approaches infinity. To get started, let's look at the given sequence:
an = {3/4, 1/8, -1/2, -9/8, ...}
To check for convergence, we need to examine if the terms of the sequence approach a specific value as n increases. One way to do this is by looking at the limit of the sequence as n approaches infinity.
1. Limit of the Sequence:
To find the limit, we consider the term-to-term behavior of the sequence. Notice that each term is obtained by multiplying the previous term by -3/2:
a1 = 3/4
a2 = (3/4) * (-3/2) = -9/8
a3 = (-9/8) * (-3/2) = 27/16
a4 = (27/16) * (-3/2) = -81/32
From these calculations, we can observe the pattern. As n increases, each term becomes the previous term multiplied by -3/2:
So, we can generalize the nth term of the sequence as follows:
an = (3/4) * (-3/2)^(n-1)
Now, as n approaches infinity, the term (-3/2)^(n-1) will tend to 0 if |(-3/2)| < 1. In this case, |-3/2| = 3/2, which is greater than 1.
Since the absolute value of the constant term (-3/2) is greater than 1, the term (-3/2)^(n-1) will not tend to zero. Therefore, the sequence does not converge.
2. Associated Series:
The associated series is obtained by summing up all the terms of the sequence. For the given sequence, the associated series is:
3/4 + 1/8 - 1/2 - 9/8 + ...
To determine if this series converges or diverges, we need to evaluate whether the sequence of partial sums converges or diverges. The partial sum of the series after n terms, Sn, is defined as:
Sn = a1 + a2 + ... + an
For our case, the partial sum after n terms is:
Sn = (3/4) + (1/8) + (-1/2) + ... + (-3/2)^(n-1)
Since we have already established that the sequence does not converge, the terms of the series do not approach a specific value as n increases. Therefore, the associated series also diverges.
In summary, the sequence an does not converge, and, consequently, the associated series 3/4 + 1/8 - 1/2 - 9/8 + ... also diverges.