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One way is:

var('k')
assume(x>-1)
assume(x<1)
(4*sum(x^(2*k), k, 0, oo)+sum(x^(2*k+1), k, 0, oo)/4).combine()

-1/4*(x + 16)/(x^2 - 1)

One way is:

var('k')
assume(x>-1)
assume(x<1)
(4*sum(x^(2*k), k, 0, oo)+sum(x^(2*k+1), k, 0, oo)/4).combine()

-1/4*(x + 16)/(x^2 - 1)

var('x k')
assume(k,'integer')
w(k)=4^((-1)^k)*x^k
a(k)=w(k=2*k).full_simplify()
b(k)=w(k=2*k+1).full_simplify();b
(sum(a(k),k,0,oo)+sum(b(k),k,0,oo)).combine()

-1/4*(x + 16)/(x^2 - 1)

One way is:

var('k')
assume(x>-1)
assume(x<1)
(4*sum(x^(2*k), k, 0, oo)+sum(x^(2*k+1), k, 0, oo)/4).combine()

-1/4*(x + 16)/(x^2 - 1)

A more automatic version:

var('x k')
assume(k,'integer')
w(k)=4^((-1)^k)*x^k
a(k)=w(k=2*k).full_simplify()
b(k)=w(k=2*k+1).full_simplify();b
(sum(a(k),k,0,oo)+sum(b(k),k,0,oo)).combine()

-1/4*(x + 16)/(x^2 - 1)

One way is:

var('k')
assume(x>-1)
assume(x<1)
(4*sum(x^(2*k), k, 0, oo)+sum(x^(2*k+1), k, 0, oo)/4).combine()

-1/4*(x + 16)/(x^2 - 1)

A more automatic version:

var('x k')
assume(k,'integer')
w(k)=4^((-1)^k)*x^k
a(k)=w(k=2*k).full_simplify()
b(k)=w(k=2*k+1).full_simplify();b
b(k)=w(k=2*k+1).full_simplify()
(sum(a(k),k,0,oo)+sum(b(k),k,0,oo)).combine()

-1/4*(x + 16)/(x^2 - 1)

(assuming that both series converge)