section 10.10 – the binomial series pascal’s triangle

Section 10.10 – The Binomial Series Pascal’s Triangle

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Section 10.10 – The Binomial Series

Pascal’s Triangle

http://en.wikipedia.org/wiki/Pascal's_triangle

Section 10.10 – The Binomial Series

Pascal’s Triangle

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&docid=iPki-eCXkzfYoM&tbnid=-VIjFuaP2MTlDM:&ved=0CAUQjRw&url=http://soundsmartabout.com/pascals-triangle-and-the-mathematics-of-fraternal-twins/&ei=fERfUunrJYm28wT6moHYCQ&bvm=bv.54176721,d.eWU&psig=AFQjCNFz5oUnePk4L1Dw7yyRfimYDfH3cQ&ust=1382061502516183

Section 10.10 – The Binomial Series

(𝑥+𝑎 )𝑛=(𝑛0 )𝑥𝑛+(𝑛1 )𝑎𝑥𝑛−1+(𝑛2 )𝑎2𝑥𝑛−2+(𝑛3 )𝑎3 𝑥𝑛−3⋯(𝑛𝑛)𝑎𝑛

(𝑥+𝑎 )𝑛=∑𝑖=0

𝑛

(𝑛𝑖 )𝑎𝑖𝑥𝑛− 𝑖 h𝑤 𝑒𝑟𝑒 (𝑛𝑖 )= 𝑛 !𝑖 ! (𝑛−𝑖 )!

The Binomial Theorem

(2 𝑥−3 )4=¿

¿1 ∙16 𝑥4+4 (−3 ) (2 𝑥 )3+6 (−3 )2 (2 𝑥 )2+4 (−3 )3 (2𝑥 )1+1 (−3 )4 (2𝑥 )0

¿16 𝑥4−96 𝑥3+216 𝑥2−216 x+81

Section 10.10 – The Binomial Series

(𝑥+𝑎 )𝑛=(𝑛0 )𝑥𝑛+(𝑛1 )𝑎𝑥𝑛−1+(𝑛2 )𝑎2𝑥𝑛−2+(𝑛3 )𝑎3 𝑥𝑛−3⋯(𝑛𝑛)𝑎𝑛

(𝑥+𝑎 )𝑛=∑𝑖=0

𝑛

(𝑛𝑖 )𝑎𝑖𝑥𝑛− 𝑖 h𝑤 𝑒𝑟𝑒(𝑛𝑖 )= 𝑛 !𝑖 ! (𝑛−𝑖 )!

The Binomial Theorem

(𝑥3+2 )3=¿

(30)(𝑥3 )3+(31)2 (𝑥3 )3−1+(32)22 (𝑥3 )3−2+(33)23 (𝑥3 )3−3

¿1 ∙𝑥9+3 ∙2 (𝑥3 )2+3 ∙4 (𝑥3 )1+1∙8 (𝑥3 )0

¿ 𝑥9+6 𝑥6+12𝑥3+8

Section 10.10 – The Binomial Series

The Binomial Series

The Binomial series is similar to the Binomial Theorem. The difference between the two is that the Binomial series is infinite.

Section 10.10 – The Binomial Series

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