Question

find the period of f(x)=sinx+sin^2x+sin^3x+.....................+sin^2020x

Answer 1

2π is Period of f(x) = sinx + sin²x + sin³x + ......... ............ + sin²⁰²⁰x  is 2π

Step-by-step explanation:

f(x)=sinx+sin²x+sin³x+.....................+sin²⁰²⁰x

this is an GP

First term a  = Sinx

Common Ratio = Sinx

number of terms  =  2020

Sum of an GP = a(1 - rⁿ )/(1 - r)

= Sinx( 1 - Sin²⁰²⁰x   )/(1 - Sinx )

Let check for π/2

then f(x + π/2) = (-Cosx)( 1 - Cos²⁰²⁰x )/(1 +Cosx)  ≠ f(x)

Let check for π

then f(x + π ) = (-Sinx)( 1 - Sin²⁰²⁰x)/(1 + Sinx)    ≠ f(x)

Let check for 2π

then f(x + 2π ) = ( Sinx)( 1 - Sin²⁰²⁰x)/(1 -Sinx) = f(x)

Hence Period of f(x)=sinx+sin²x+sin³x+.....................+sin²⁰²⁰x  is 2π

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