Calculating sum of fractions without calculator

Question

Calculate $$\dfrac{1}{25}+\dfrac{1}{26}+\dfrac{1}{27}+\dfrac{1}{28}+\dfrac{1}{29}=?$$ without using calculator

Update: Is there any shortcut to determine the sum of this type of fractional series?

Answer 1

$$S=\dfrac{1}{25}+\dfrac{1}{26}+\dfrac{1}{27}+\dfrac{1}{28}+\dfrac{1}{29}=\frac 1 {25}\left(1+\sum_{n=1}^4\frac{1}{1+\frac n{25} }\right)$$ Now, by Taylor $$\frac 1{1+\frac n{25}}=1-\frac{n}{25}+\frac{n^2}{625}-\frac{n^3}{15625}+O\left(n^4\right)$$ $$\sum_{n=1}^4\frac{1}{1+\frac n{25} }=\frac{2276}{625}$$ $$S=\frac 1 {25}\left(1+\frac{2276}{625}\right)=\frac{2901}{15625}=0.1856640$$ while the exact value is $$S=\frac{1323137}{7125300}\sim 0.1856956$$ Then a relative error of $0.017$%.

Done by hand

Edit

Another solution $$S=\dfrac{1}{25}+\dfrac{1}{26}+\dfrac{1}{27}+\dfrac{1}{28}+\dfrac{1}{29}$$ $$S=\dfrac{1}{27-2}+\dfrac{1}{27-1}+\dfrac{1}{27}+\dfrac{1}{27+1}+\dfrac{1}{27+2}$$ Factor $$27S-1=\frac 1 {1-\frac 2 {27}}+\frac 1 {1+\frac 2 {27}}+\frac 1 {1-\frac 1 {27}}+\frac 1 {1+\frac 1 {27}}$$ Now $$\frac 1{1-\epsilon}+\frac 1{1+\epsilon}=2+2\epsilon^2+O\left(\epsilon^4\right)$$ $$27S-1=2+\frac{8}{729}+2+\frac{2}{729}=\frac{2926}{729}\implies S=\frac{3655}{19683}\sim 0.1856932$$ which is much better (but I prefer to divide by $25$ rather by $27$ even if $27=3^3$).

Still done by hand

Answer 2

I did all of the following without writing any "multiplication algorithms" down, and it's based on the assumption that one can "add two numbers" and "multiply any number with a one-digit number" quickly, and we wrote down all previous results for later use.

Notice that $$27\times29=87\times9=783$$

First we calculate $$25\times26\times27\times28\times29=25\times8\times7\times13\times783$$ $$=200\times(90\times783+783)=200\times(78300+783-7830)=200\times(79253-8000)$$ $$=200\times71253=14250600$$

Next we calculate $$26\times27\times28\times29=8\times71253=570024$$

Next we calculate $$25\times27\times 28\times 29 = 25\times4\times7\times783=100\times5481=548100$$

Next we calculate $$25\times26\times28\times29=25\times28\times(27-1)\times29$$ $$25\times4\times7\times(783-29)=100\times7\times754=527800$$

Next we calculate $$25\times26\times27\times29=50\times13\times783=700\times783-50\times783=548100-39150=508950$$

Finally we calculate $$25\times26\times27\times28=25\times28\times27\times(29-3)$$ $$=25\times4\times7\times(783-81)=700\times702=491400$$

Now addition time,

$$570024+548100+527800+508950+491400=2646274$$

Now the final result,

$$2646274\over 14250600$$