ExamJEE Classes

Frequently used Series Expansions

A series expansion is a representation of a particular function as a sum of powers in one of its variables, or by a sum of powers of another (usually elementary) function f(x).

Here are series expansions for a number of common functions.

  1. $latex \displaystyle { e }^{ x }=1+\frac { x }{ 1! } +\frac { { x }^{ 2 } }{ 2! } +\frac { { x }^{ 3 } }{ 3! } +…… &s=1$
  2. $latex \displaystyle { a }^{ x }=1+\frac { x\log { a } }{ 1! } +\frac { { x }^{ 2 }{ \left( loga \right) }^{ 2 } }{ 2! } +…..\qquad where\quad a\quad \epsilon \quad { R }^{ + } &s=1$
  3. $latex \displaystyle { \left( 1+x \right) }^{ 2 }=1+\frac { nx }{ 1! } +\frac { n\left( n-1 \right) { x }^{ 2 } }{ 2! } +\frac { n\left( n-1 \right) \left( n-2 \right) { x }^{ 3 } }{ 3! } +…..n\quad \epsilon \quad R\quad and\quad \left| x \right| \textless 1 &s=1$
  4. $latex \displaystyle \log { (1+x) } =x-\frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 3 } }{ 3 } -\frac { { x }^{ 4 } }{ 4 } +……\qquad where,-1\le x\le 1 &s=1$
  5. $latex \displaystyle \frac { { x }^{ n }-{ a }^{ n } }{ x-a } ={ x }^{ n-1 }+{ x }^{ n-2 }a+{ x }^{ n-3 }{ a }^{ 2 }+……+{ a }^{ n-1 } &s=1$
  6. $latex \displaystyle { \left( 1+x \right) }^{ \frac { 1 }{ x } }=e\left( 1-\frac { x }{ 2 } +\frac { 11{ x }^{ 2 } }{ 24 } +…. \right) &s=1$
  7. $latex \displaystyle \sin { x } =x-\frac { { x }^{ 3 } }{ 3! } +\frac { { x }^{ 5 } }{ 5! } -\frac { { x }^{ 7 } }{ 7! } +….. &s=1$
  8. $latex \displaystyle \cos { x } =1-\frac { { x }^{ 2 } }{ 2! } +\frac { { x }^{ 4 } }{ 4! } -\frac { { x }^{ 6 } }{ 6! } +….. &s=1$
  9. $latex \displaystyle \tan { x } =x+\frac { { x }^{ 3 } }{ 3 } +\frac { 2 }{ 15 } { x }^{ 5 }+….. &s=1$
  10. $latex \displaystyle \sin ^{ -1 }{ x } =x+\frac { { 1 }^{ 2 } }{ 3! } { x }^{ 3 }+\frac { { 1 }^{ 2 }{ 3 }^{ 2 } }{ 5! } { x }^{ 5 }+\frac { { 1 }^{ 2 }{ 3 }^{ 2 }{ 5 }^{ 2 } }{ 7! } { x }^{ 7 }+….. &s=1$
  11. $latex \displaystyle \tan ^{ -1 }{ x } =x-\frac { { x }^{ 3 } }{ 3 } +\frac { { x }^{ 5 } }{ 5 } ….. &s=1$
  12. $latex \displaystyle { \left( 1+x \right) }^{ \frac { 1 }{ x } }=\left\{ 1-\frac { x }{ 2 } +\frac { 11 }{ 24 } { x }^{ 2 }+……. \right\} &s=1 $
April 18, 2019
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