202109101417 Exercises 1.1.A (Q5)

By equation (1.1), $\pi =4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots )$ , where the $n^{\textrm{th}}$ term in the sum inside the parenthesis is $\frac{(-1)^{n+1}}{2n-1}$ (starting at $n=1$ ). So the first approximation of $\pi$ using this formula is $\pi\approx 4(1)=4.0$ , and the second approximation is $\pi\approx 4(1-\frac{1}{3})=8/3\approx 2.66667$ . Continue like this until two consecutive approximations have $3$ as the first digit before the decimal point. How many terms in the sum did this require? Be careful with rounding off in the approximations.

Attempts.

$1^{\textrm{st}}$ approximation:

$\pi\approx 4(1)=4.0$

$2^{\textrm{nd}}$ approximation:

$\pi\approx 4(1-\frac{1}{3})=\frac{8}{3}\approx 2.66667$

$3^{\textrm{rd}}$ approximation:

$\pi\approx 4(1-\frac{1}{3}+\frac{1}{5})=\frac{52}{15}\approx 3.466667\quad (\textrm{5 d.p.})$

$4^{\textrm{th}}$ approximation:

$\pi\approx 4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7})=\frac{304}{105}\approx 2.89523\quad (\textrm{5 d.p.})$

$5^{\textrm{th}}$ approximation:

$\pi\approx 4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9})=\frac{1052}{315}\approx 3.33968\quad (\textrm{5 d.p.})$

$6^{\textrm{th}}$ approximation:

$\pi\approx 4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11})=\frac{10312}{3465}\approx 2.97605\quad (\textrm{5 d.p.})$

$7^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}\bigg) \\ & =\frac{147916}{45045} \\ & \approx 3.28374\quad (\textrm{5 d.p.}) \end{aligned}$

$8^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}\bigg) \\ & =\frac{135904}{45045} \\ & \approx 3.01707\quad (\textrm{5 d.p.}) \end{aligned}$

$9^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}\bigg) \\ & =\frac{2490548}{765765} \\ & \approx 3.25237\quad (\textrm{5 d.p.}) \end{aligned}$

$10^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}\bigg) \\ & =\frac{44257352}{14549535} \\ & \approx 3.04184\quad (\textrm{5 d.p.}) \end{aligned}$

$11^{\textrm{th}}$ approximation:

$12^{\textrm{th}}$ approximation:

$13^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}\bigg) \\ & =\frac{5385020324}{1673196525} \\ & \approx 3.21840\quad (\textrm{5 d.p.}) \end{aligned}$

$14^{\textrm{th}}$ approximation:

$15^{\textrm{th}}$ approximation:

$16^{\textrm{th}}$ approximation:

$17^{\textrm{th}}$ approximation:

$18^{\textrm{th}}$ approximation:

$19^{\textrm{th}}$ approximation:

$20^{\textrm{th}}$ approximation:

$21^{\textrm{st}}$ approximation:

$22^{\textrm{nd}}$ approximation:

$23^{\textrm{rd}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\bigg) \\ & =\frac{937558224301357108}{294362129962575675} \\ & \approx 3.18505\quad (\textrm{5 d.p.}) \end{aligned}$

$24^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}\bigg) \\ & =\frac{42887788022313481376}{13835020108241056725} \\ & \approx 3.09994\quad (\textrm{5 d.p.}) \end{aligned}$

$25^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}\bigg) \\ & =\frac{308120241932332116332}{96845140757687397075} \\ & \approx 3.18158\quad (\textrm{5 d.p.}) \end{aligned}$

$26^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}\bigg) \\ & =\frac{300524544618003693032}{96845140757687397075} \\ & \approx 3.10315\quad (\textrm{5 d.p.}) \end{aligned}$

$27^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}+\frac{1}{53}\bigg) \\ & =\frac{16315181427784945318996}{5132792460157432044975} \\ & \approx 3.17862\quad (\textrm{5 d.p.}) \end{aligned}$

$28^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}+\frac{1}{53}-\frac{1}{55}\bigg) \\ & =\frac{15941887430682586624816}{5132792460157432044975} \\ & \approx 3.10589\quad (\textrm{5 d.p.}) \end{aligned}$

$29^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}+\frac{1}{53}-\frac{1}{55}+\frac{1}{57}\bigg) \\ & =\frac{16302083392798897645516}{5132792460157432044975} \\ & \approx 3.17607\quad (\textrm{5 d.p.}) \end{aligned}$

$30^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}+\frac{1}{53}-\frac{1}{55}+\frac{1}{57}-\frac{1}{59}\bigg) \\ & =\frac{941291750334505232905544}{302834755149288490653525} \\ & \approx 3.10827\quad (\textrm{5 d.p.}) \end{aligned}$

$31^{\textrm{st}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}+\frac{1}{53}-\frac{1}{55}+\frac{1}{57}-\frac{1}{59}+\frac{1}{61}\bigg) \\ & =\frac{58630135791001973169852284}{18472920064106597929865025} \\ & \approx 3.17384\quad (\textrm{5 d.p.}) \end{aligned}$

$32^{\textrm{nd}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}+\frac{1}{53}-\frac{1}{55}+\frac{1}{57}-\frac{1}{59}+\frac{1}{61}-\frac{1}{63}\bigg) \\ & =\frac{57457251977407903460019584}{18472920064106597929865025} \\ & \approx 3.11035\quad (\textrm{5 d.p.}) \end{aligned}$

$33^{\textrm{rd}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}+\frac{1}{53}-\frac{1}{55}+\frac{1}{57}-\frac{1}{59}+\frac{1}{61}-\frac{1}{63}+\frac{1}{65}\bigg) \\ & =\frac{4507234389098153984166548}{1420993851085122917681925} \\ & \approx 3.17189\quad (\textrm{5 d.p.}) \end{aligned}$

$34^{\textrm{th}}$ approximation:

$\begin{aligned} \pi & \approx 4\bigg( 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\frac{1}{17}-\frac{1}{19}+\frac{1}{21}-\frac{1}{23} \\ & \qquad\quad +\frac{1}{25}-\frac{1}{27}+\frac{1}{29}-\frac{1}{31}+\frac{1}{33}-\frac{1}{35}+\frac{1}{37}-\frac{1}{39}+\frac{1}{41}-\frac{1}{43}+\frac{1}{45}\\ & \qquad\qquad\quad -\frac{1}{47}+\frac{1}{49}-\frac{1}{51}+\frac{1}{53}-\frac{1}{55}+\frac{1}{57}-\frac{1}{59}+\frac{1}{61}-\frac{1}{63}+\frac{1}{65}-\frac{1}{67}\bigg) \\ & =\frac{296300728665235825268431016}{95206588022703235484688975} \\ & \approx 3.11219\quad (\textrm{5 d.p.}) \end{aligned}$

Summing without aim, I forgot my purpose. Where am I?

(discontinued)

(refreshed)

Please scroll up to the $7^{\textrm{th}}$ and $8^{\textrm{th}}$ approximation.

This required seven or eight terms in the sum for having $3$ as the first digit before the decimal point.

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