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# The limiting sum of the infinite series [m][fraction]1/2[/fraction] +

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Joined: 23 Feb 2015
Posts: 2008
The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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Updated on: 22 May 2020, 13:12
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Question Stats:

75% (02:32) correct 25% (05:09) wrong based on 4 sessions

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The limiting sum of the infinite series $$\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \frac{7}{16} + \frac{9}{32}$$+ ... whose $$n$$th term is $$\frac{2n-1}{2^n}$$ is:
A) $$\frac{5}{2}$$
B) 3
C) $$\frac{7}{2}$$
D) $$\frac{15}{4}$$
E) 4

Source: Quantum GMAT

Originally posted by TheUltimateWinner on 22 May 2020, 08:24.
Last edited by TheUltimateWinner on 22 May 2020, 13:12, edited 1 time in total.
Tutor
Joined: 16 May 2019
Posts: 989
Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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28 May 2020, 14:27
2
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..

Ah, the spoiler that I sometimes cannot get to work. I am not sure if it is a browser issue or a computer issue, but sometimes I click on the spoiler or the OA/OE and get nothing. It worked this time, so I feel foolish. (Oh well.) I would like to hit a 51 in Quant myself. Maybe later this year, once matters settle down, I will put myself to the test and get a Verified Score to post... hopefully with a pair of 51s to show for the effort. Time will tell.

- Andrew
Tutor
Joined: 16 May 2019
Posts: 989
Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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22 May 2020, 12:58
1
The limiting sum of the infinite series $$\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \frac{7}{16} + \frac{9}{31}$$+ ... whose $$n$$th term is $$\frac{2n-1}{2^n}$$ is:
A) $$\frac{5}{2}$$
B) 3
C) $$\frac{7}{2}$$
D) $$\frac{15}{4}$$
E) 4

Source: Quantum GMAT

Hello, Asad. Thank you for posting this question, but I think the fifth term should be $$\frac{9}{32}$$ instead. It is not that I think anyone solving the question would be flummoxed by the change, but for the sake of consistency with the established rule, I thought I would bring the issue to your attention.

- Andrew
Tutor
Joined: 16 May 2019
Posts: 989
The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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28 May 2020, 12:40
1
Hello again, Asad. I wanted to outline my approach to the question, in case it differs from your own or that of any onlooker. I typically favor a more intuitive approach than a rigorous mathematical approach. In the question at hand, I started by converting all the fractions to thirty-seconds and adding:

$$\frac{16}{32}+\frac{24}{32}+\frac{20}{32}+\frac{14}{32}+\frac{9}{32}=\frac{83}{32}$$

Now, our next fraction would be in sixty-fourths, per the pattern, and the numerator would be 11. To put this value together with the earlier sum in a meaningful way, I converted the earlier answer to sixty-fourths:

$$\frac{11}{64}+\frac{166}{64}=\frac{177}{64}$$

At this point, we have a decimal value that would lie between 2 and 3, but closer to the 3 end, since

$$64*3=192$$

Looking at the answer choices, we can eliminate (A) since that is already too low. How do we know?

$$64*2+32<177$$

We can also appreciate, without doing more math, that the sum will grow in smaller and smaller increments, and our last term was already about one-sixth. Thus, it appears as though our sum will converge on 3. Still, for the inner fact-checker, we could go on. In terms of sixty-fourths, we have 15 more to add in the numerator (to get from 177 to 192), and that is going to take multiple steps. The next two terms would give us

$$\frac{13}{128}+\frac{15}{256}$$

or, in terms of sixty-fourths, the numerators would be

$$\frac{13}{2}+\frac{15}{4}=\frac{41}{4}$$

Closer still, as we would have just 4.75 more to go to get our numerator to 192. But to be honest, I see no way that, in keeping with this pattern, the numerator will increase to 192 and another 32 (half of 64) to get us up to answer (C). Although I would have selected my answer already, if I had not before, I would feel confident choosing (B), confirming, and moving on to the next challenge.

Thank you for sharing the question. What is the source, if you do not mind my asking?

- Andrew
SVP
Joined: 23 Feb 2015
Posts: 2008
The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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28 May 2020, 14:46
1
MentorTutoring wrote:
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..

Ah, the spoiler that I sometimes cannot get to work. I am not sure if it is a browser issue or a computer issue, but sometimes I click on the spoiler or the OA/OE and get nothing. It worked this time, so I feel foolish. (Oh well.) I would like to hit a 51 in Quant myself. Maybe later this year, once matters settle down, I will put myself to the test and get a Verified Score to post... hopefully with a pair of 51s to show for the effort. Time will tell.

- Andrew

good luck for you so that you can get q51! in the mean time you can see the explanation from the following link (if you need..)

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Tutor
Joined: 16 May 2019
Posts: 989
Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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28 May 2020, 15:09
1
MentorTutoring wrote:
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..

Ah, the spoiler that I sometimes cannot get to work. I am not sure if it is a browser issue or a computer issue, but sometimes I click on the spoiler or the OA/OE and get nothing. It worked this time, so I feel foolish. (Oh well.) I would like to hit a 51 in Quant myself. Maybe later this year, once matters settle down, I will put myself to the test and get a Verified Score to post... hopefully with a pair of 51s to show for the effort. Time will tell.

- Andrew

good luck for you so that you can get q51! in the mean time you can see the explanation from the following link (if you need..)

Thank you for the well wishes, as well as for the link. (I am not on Facebook, so I would not have found anything on my own that way.) Kudos from me for all the help.

- Andrew
SVP
Joined: 23 Feb 2015
Posts: 2008
The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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22 May 2020, 13:15
MentorTutoring wrote:
The limiting sum of the infinite series $$\frac{1}{2} + \frac{3}{4} + \frac{5}{8} + \frac{7}{16} + \frac{9}{31}$$+ ... whose $$n$$th term is $$\frac{2n-1}{2^n}$$ is:
A) $$\frac{5}{2}$$
B) 3
C) $$\frac{7}{2}$$
D) $$\frac{15}{4}$$
E) 4

Source: Quantum GMAT

Hello, Asad. Thank you for posting this question, but I think the fifth term should be $$\frac{9}{32}$$ instead. It is not that I think anyone solving the question would be flummoxed by the change, but for the sake of consistency with the established rule, I thought I would bring the issue to your attention.

- Andrew

MentorTutoring
You're right. Edited.....
Thank you...
SVP
Joined: 23 Feb 2015
Posts: 2008
The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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28 May 2020, 12:55
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..
SVP
Joined: 23 Feb 2015
Posts: 2008
Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +  [#permalink]

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28 May 2020, 15:18
MentorTutoring wrote:
MentorTutoring wrote:
The source is 'Quantum GMAT'... Already source mentioned in the spoiler. This question has been designed for them who are targeting Q51! Thanks..

Ah, the spoiler that I sometimes cannot get to work. I am not sure if it is a browser issue or a computer issue, but sometimes I click on the spoiler or the OA/OE and get nothing. It worked this time, so I feel foolish. (Oh well.) I would like to hit a 51 in Quant myself. Maybe later this year, once matters settle down, I will put myself to the test and get a Verified Score to post... hopefully with a pair of 51s to show for the effort. Time will tell.

- Andrew

good luck for you so that you can get q51! in the mean time you can see the explanation from the following link (if you need..)

Thank you for the well wishes, as well as for the link. (I am not on Facebook, so I would not have found anything on my own that way.) Kudos from me for all the help.

- Andrew

I think you should not have any facebook ID to watch that video! if you click on this link it'll be automatically started..
Re: The limiting sum of the infinite series [m][fraction]1/2[/fraction] +   [#permalink] 28 May 2020, 15:18