This is a geometric series. The sum of geometric series a/(1-r) for |r|<1. f(x) is a convergent geometric series with sum 1+x^2. Show how f(x) can be defined for different values o... This is a geometric series. The sum of geometric series a/(1-r) for |r|<1. f(x) is a convergent geometric series with sum 1+x^2. Show how f(x) can be defined for different values of x and discuss its potential discontinuities. Read more

Steps to Solve

1. Understanding Geometric Series A geometric series is defined with a common ratio $r$. The sum can be expressed as: $$ S_n = a + ar + ar^2 + ar^3 + \ldots + ar^{n-1} $$

For $|r| < 1$, the infinite sum of the series converges to: $$ S = \frac{a}{1 - r} $$

2. Defining the Function ( f(x) ) We want to consider ( f(x) ) as a series: $$ f(x) = \sum_{n=0}^{\infty} a_n $$ with convergence criteria based on the nature of terms at ( x \neq 0 ).

3. Examining Rational and Irrational Inputs For rational values of ( x ): Let ( x = \frac{p}{q} ). Then, we evaluate if the sequence ( m \cdot x ) for integers ( m ) leads to periodic behavior influencing ( f(x) ).

4. Limits Involving Cosine Function Consider: $$ f_m(x) = \lim_{n \to \infty} \cos(m\pi x) 2^{-n} $$ If ( mx ) is an integer, this limit suggests ( f_m(x) = 1 ). If not, evaluate the limit behavior as ( n \to \infty ).

5. Continuous Limit and Discontinuity Assess the computed ( f(x) ) overall:

• If derived limits converge to points resulting in consistent output, ( f(x) ) is continuous.
• If derived outputs differ for rational and irrational inputs, a discontinuity is established.