1+1/2+1/3+.....+1/16 is

Question

Qs.  1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{16}  is

A) between 1 and 2
B) between 2 and 3
C) between 3 and 4
D) between 4 and 5
E) between 5 and 6

KarishmaB · Accepted Answer

I don't think the question is appropriate for GMAT until and unless they want you to do lots of calculations and then approximate. You are not expected to find the sum of harmonic mean.

The sum of harmonic series is approximated by comparison.

S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + ...

T = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{16} + ...

Every term of S is greater than or equal to corresponding term of T.

T = 1 + \frac{1}{2} +   +   +

T = 1 + \frac{1}{2} +   +   +   = 3

So S must be more than 3.

Now note that max diff between two corresponding terms of S and T is 1/3 - 1/4 = 1/12. There are only 11 terms in S which are more than their corresponding terms in T. So even if we account for all the difference as 11*(1/12) = 11/12 and add it to 3, we will still not get 4.

Hence S must lie between 3 and 4.

Narenn · Answer

Sum =  1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}...........

Sum = 1 + 0.5 + 0.33 + 0.25 + 0.20 + 0.16 + 0.14 + 0.12 + 0.11 + ..............

Sum = 2.81 + .........

Each of 1/10 and 1/11 would add approximately 0.10 to the above sum, so we can infer that the sum will definitely be greater than 3 but would not go beyond 4

Choice C

GGMAT760 · Answer

Hi I did not get, how did you write T terms??
Can anyone please explain? Or is there another way to do this problem???

KarishmaB · Answer

This is a standard method of comparing the sum of harmonic series. We compare the sum with another series in which the terms are easy to add and the sum is easy to find. That other series is T. Compare the corresponding terms of S and T - i.e. first term of S with first term of T, second term of S with second term of T and so on. Every term of S will be greater than or equal to the corresponding term of T.

From where do we get T? It has been obtained by mathematicians using the properties of numbers and their powers. If you don't know it, it's highly unlikely that you will be able to think of it in the exam. GMAT doesn't expect you to. The other method is using approximate values of the fractions and adding them but that's way too cumbersome in my opinion. GMAT doesn't do cumbersome.

But if you do understand the method used by me above, as an intellectual exercise, try solving for S = 1 + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + ... + 1/27

arunspanda · Answer

To find : S = 1 + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + ... + 1/27

Let $S' = (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} +\frac{1}{6} +\frac{1}{7} +\frac{1}{8}+.....+ \frac{1}{32} )$

Or, $S' = ( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + ... + \frac{1}{32}) + (\frac{1}{3} +\frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + ... + \frac{1}{27}+\frac{1}{29}+\frac{1}{31})$

Or, $S' = 1 + (\frac{1}{2} (1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} +\frac{1}{10} + ... + \frac{1}{16})) + (\frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + ... + \frac{1}{27})+(\frac{1}{29}+\frac{1}{31})$

Or,$S' > ( \frac{1}{2}( 1+ 4(\frac{1}{2})) + (1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{9} + \frac{1}{11} + ... + \frac{1}{27})+(\frac{1}{29}+\frac{1}{31})$

Or, $S' > \frac{3}{2} + S + (\frac{1}{29}+\frac{1}{31})$

Or, $S' > \frac{3}{2} + S + \frac{1}{15}$ [Since, $\frac{1}{29} +\frac{1}{31} >\frac{2}{30} or \frac{1}{15}$]

Or, $S < S' - \frac{3}{2} -\frac{1}{15}$

Or, $S < 1 + \frac{5}{2}-\frac{3}{2}-\frac{1}{15}$ [ Since, $S ' > 1 + 5(\frac{1}{2})$ ]

Or. $S < 2 - \frac{1}{15}$

Or, $S < \frac{29}{15}$

This calculated value is incorrect, as S > 2. Where have I gone wrong?

bazc · Answer

If this comes on the GMAT, I'd prefer going the decimal addition way. Decimals till 1/10 we should know for the GMAT in my opinion. Quick tenths digits addition till 1/10 should give the required range.

stonecold · Answer

Outstanding Question.
I picked the wrong Answer as a result of which this is going in my error log.

Here is what i will do the next time i see this Question =>

Let n=1+1/2+1/3+...

No need to know the complicated harmonic sum.
n=>Add the terms
1
0.5
0.33
0.25
0.20
0.16
0.14
0.12
0.11
0.10
0.09
and some more terms

Add them up we get 3
So the sum must be greater than 3.

Abhishek009  You have any tricks for this one ?

broall · Answer

Here is my solution for this one. $\begin{split} A&=1+(\frac{1}{2}+\frac{1}{3})+(\frac{1}{4}+\frac{1}{5})+...+(\frac{1}{14}+\frac{1}{15})+\frac{1}{16}\ &<1+(\frac{1}{2}+\frac{1}{2})+(\frac{1}{4}+\frac{1}{4})+...+(\frac{1}{14}+\frac{1}{14})+\frac{1}{16}\ &=1+\frac{2}{2}+\frac{2}{4} +\frac{2}{6} +...+ \frac{2}{14}+\frac{1}{16}\ &=1+1+(\frac{1}{2}+\frac{1}{3})+(\frac{1}{4}+\frac{1}{5})+(\frac{1}{6}+\frac{1}{7})+\frac{1}{16}\ &<1+1+(\frac{1}{2}+\frac{1}{2})+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{6}+\frac{1}{6})+\frac{1}{16}\ &=2+\frac{2}{2}+\frac{2}{4}+\frac{2}{6}+\frac{1}{16}\ &=2+1+\frac{1}{2}+\frac{1}{3}+\frac{1}{16}\ &=3 + \frac{43}{48}<4 \end{split}$ $\begin{split} A&=1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+...+(\frac{1}{15}+\frac{1}{16})\ &>1+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+...+(\frac{1}{16}+\frac{1}{16})\ &=1+\frac{1}{2}+\frac{2}{4}+...+\frac{2}{14}+\frac{2}{16}\ &=(1+\frac{1}{2}+\frac{1}{2})+(\frac{1}{3}+\frac{1}{4})+...+(\frac{1}{7}+\frac{1}{8})\ &>2+(\frac{1}{4}+\frac{1}{4})+...+(\frac{1}{8}+\frac{1}{8})\ &=2+\frac{2}{4}+\frac{2}{6}+\frac{2}{8}\ &=2+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\ &=2+\frac{13}{12}>3 \end{split}$ Hence $3 < A < 4$, the answer is C The solution for this type of question requires a lot of time, so it isn't the best way to solve this one under 2 minutes during take actual GMAT test. We could solve for the general problem like this one: $B=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ also, calculating approximate value of B is rather tough.

broall · Answer

Here is the solution for the general problem. This one is out of scope of GMAT, but I still post here for anyone who wants to understand the concept.
\

We have
\

Since $\frac{5\times 2^n+6}{6\times 2^n} < 1 \iff 6 \leq 2^n$, so we have $A<n$.

\

Finally, we have $1+\frac{n}{2} < A < n$

With $n=4 \implies 3 < A< 4$

Phew, quite hard one.

SeregaP · Answer

I got there just by calculations: we can estimate that the number is more than 3

BALAGURU · Answer

fist 5 terms ( 1, 1/2. 1/3, 1/4, 1/5) ~ 2.25
next 5 terms ( 1/ 6 --- to 1/10) ~ 0.17 to 0.1 ==> max value can be <= .7
next 6 terms ( 1/1 1 ---- 1/ 16) ~ 0.09 to 0.07 ==> max value can be <= 0.5

hence total ~ >= 2.25 + .7 + .5 ==> ~ 3.3 +
So 3 to 4

CEdward · Answer

Super time consuming. I semi-brute forced it.

Sum everything from 1 to 1/10. That gives you ~2.9

Next, since we have 10 fractions left, let's assume they are all equal to 1/10.

1/10 x 10 = 1

2.9 + 1 = 3.9

Answer is C. (We know in fact that all the fractions after 1/10 are less than 0.10, so the answer must be b/w 3 and 4)

errorlogger · Answer

Hi experts, need your feedback on this.

1 + (1/2 + 1/3 +......1/16)
1 + (Sum of 15 consecutive numbers)
Maximum of these numbers is 1/2, so the highest possible value = 1/2 * 15 = 7.5
so, 1 + 7.5 = 8.5 is ceil

Similarly, minimum of these numbers is 1/16, so least possible value = 1/16 * 15 = 1 (approx)
so, 1 + 1 = 2 is floor.

What am I doing wrong here?
 Bunuel

MBAB123 · Answer

Hey  errorlogger , you can't use that logic here. The sum of the series would be a fixed value, it can never be 7.5 and will always be more than 1, but that does not help us in anyway. Here, the answer choices are so close that you need to approximate the sum and come up with a solution. The maximum and minimum way only works if the choices are very different, but that is not the case here.

Also, you're using the wrong words here. It's not actually possible for the sum to be 7.5 or to be 2. In fact, it's actually impossible for the sum to take these values.

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1+1/2+1/3+.....+1/16 is

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