{"id":499,"date":"2019-06-29T22:03:25","date_gmt":"2019-06-29T22:03:25","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=499"},"modified":"2020-01-03T22:32:05","modified_gmt":"2020-01-03T22:32:05","slug":"taylor-made-pi","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/taylor-made-pi\/","title":{"rendered":"Taylor-made Pi"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

After learning about the\u00a0Taylor series\u00a0for 1\/(1+x) in calculus, you can find an interesting expression for Pi very easily.<\/p>\n\n\n\n

Start with<\/p>\n\n\n\n

1\/(1+w) = 1 – w + w2<\/sup>\u00a0– w3<\/sup>\u00a0+ …<\/p>\n\n\n\n

Now substitute x2<\/sup>\u00a0for w:<\/p>\n\n\n\n

1\/(1+x2<\/sup>) = 1 – x2<\/sup>\u00a0+ x4<\/sup>\u00a0– x6<\/sup>\u00a0+ …<\/p>\n\n\n\n

Then integrate both sides (from x=0 to x=y):<\/p>\n\n\n\n

arctan y = y – y3<\/sup>\/3 + y5<\/sup>\/5 – y7<\/sup>\/7 +…<\/p>\n\n\n\n

and plug in y=1, to get<\/p>\n\n\n\n

Pi\/4 = 1 – 1\/3 + 1\/5 – 1\/7 + …<\/p>\n\n\n\n

Voila!<\/p>\n\n\n\n

There are other\u00a0pi formulas that converge faster.<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
An alternate way to present this is to start with the well-known formula for Pi, and then present this as a “justification”.<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
Well, we glossed over the issue of why you can integrate the infinite series term by term, so if you wish to learn about this and more about Taylor series, this material is often covered in a fun course called\u00a0real analysis.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Taylor-made Pi.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Arthur Benjamin <\/p>\n","protected":false},"excerpt":{"rendered":"

After learning about the\u00a0Taylor series\u00a0for 1\/(1+x) in calculus, you can find an interesting expression for Pi very easily. Start with...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[57,2,20,7,72,41,30,173],"class_list":["post-499","page","type-page","status-publish","hentry","tag-advanced","tag-algebra","tag-analysis","tag-calculus","tag-geometric-series","tag-infinite-series","tag-pi-formula","tag-taylor-series"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/499","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=499"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/499\/revisions"}],"predecessor-version":[{"id":1675,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/499\/revisions\/1675"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=499"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=499"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}