Solving Problems in Differential Calculus |
From inside the book
Results 1-3 of 4
Page 91
... Binomial series . For any value of n , if | x | < 1 ( 1 + x ) = 1 + nx + 1 1 1 + nx + , n ( n − 1 ) x + ân ( n − 1 ) ( n - 2 ) x + · n ( n − 1 ) x2 + — ̧n ( n ... series . For | x $ 21 91 APPLICATIONS TO INFINITE SERIES Maclaurin's series.
... Binomial series . For any value of n , if | x | < 1 ( 1 + x ) = 1 + nx + 1 1 1 + nx + , n ( n − 1 ) x + ân ( n − 1 ) ( n - 2 ) x + · n ( n − 1 ) x2 + — ̧n ( n ... series . For | x $ 21 91 APPLICATIONS TO INFINITE SERIES Maclaurin's series.
Page 95
... series the function - 1 3 / ( 8-4x ) 2 ° We note that ( 8-4x ) −3 = 8 − 3 ( 1 − 4x ) ̃3 = 4 { 1 + ( − ‡ 4x ) } ̄3 , so , if | -4x | < 1 , i.e. , if | x | < 2 , expanding as a Binomial series gives · ↓ + { { 1 1 + + ( ( − − 3 ) 3 ) ...
... series the function - 1 3 / ( 8-4x ) 2 ° We note that ( 8-4x ) −3 = 8 − 3 ( 1 − 4x ) ̃3 = 4 { 1 + ( − ‡ 4x ) } ̄3 , so , if | -4x | < 1 , i.e. , if | x | < 2 , expanding as a Binomial series gives · ↓ + { { 1 1 + + ( ( − − 3 ) 3 ) ...
Page 96
... expansion a as power series in x , as far as the term in x3 , of √ ( 1 + 2x + 3x2 ) . Writing √ ( 1 + 2x + 3x2 ) = { 1+ ( 2x + 3x2 ) } * and expanding as a Binomial series , 1 √ / ( 1 + 2x + 3x2 ) = 1 + ¦ ( 2x + 3x2 ) + ( 2 ) ( − 1 ) ...
... expansion a as power series in x , as far as the term in x3 , of √ ( 1 + 2x + 3x2 ) . Writing √ ( 1 + 2x + 3x2 ) = { 1+ ( 2x + 3x2 ) } * and expanding as a Binomial series , 1 √ / ( 1 + 2x + 3x2 ) = 1 + ¦ ( 2x + 3x2 ) + ( 2 ) ( − 1 ) ...
Common terms and phrases
1+x² 2x sin 3x absolutely convergent ADDITIONAL EXAMPLES angle approximately Binomial series conditionally convergent constant cos² 5x cos³ cosec cosh cosh x cosh2 coth curve d²y dx2 d²y dy d²y/dx² decimal places Deduce derivative Differentiate dt dt dx dt dx dy dx2 dx dy dx dy/dx equation feet Find the stationary finite limit harmonic series Hence hyperbolic functions increases J₁ Leibnitz's Theorem logarithm Maclaurin expansion Maclaurin series maximum minimum n²)n nt+a oscillates finitely polynomial positive integer positive terms power series power series expansion prove radians radius of convergence sec² sec³ sech series diverges series is convergent sin x sin¯¹ sin¹ sin² sin³ sinh sinh2 Sketch the graph SPDC stationary points tangent tends to infinity term in x4 u₁ velocity x-sin x+cos x²+1 y₁ Ип