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Sum

Expand `2x^3+7x^2+x-1` in powers of x - 2

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#### Solution

Let `f(x) =2x^3+7x^2+x-1`

Here a = 2

`f(x) =2x^3+7x^2+x-1` | 𝒇(𝟐)=𝟒𝟓 |

`f'(x) =6x^2+14x+1` | 𝒇′(𝟐)=𝟓𝟑 |

`f''(x) =12x+14` | 𝒇′′(𝟐)=𝟑𝟖 |

𝒇′′′(𝒙)=𝒇′′′(𝟐)=𝟏𝟐

Taylor’s series is :

`f(x)=f(a)+(x-a)f'(a)+(x-a)^2/(2!)f''(a)+....`

`2x^3+7x^2+x-1=45+(x-2)53+(x-2)^2/(2!)38+(x-a)^3/(3!)12`

`2x^3+7x^2+x-1=45+53(x-2)+19(x-2)^2+2(x-2)^3`

Concept: Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ

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