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Let $\lambda_i (i=1,...,k)$ be any nonzero complex scalars and $\varphi_i(i=1,..,k)$ be any analytic self-maps of the unit disk $\mathbb{D}$. We showthat the operator $\sum_{i=1}^k\lambda_iC_{\varphi_i}$ is compact on the Blochspace $\mathcal{B}$ if and only if  $$\lim_{n\to\infty}\|\lambda_1\varphi_1^n+\lambda_2\varphi_2^n+...+\lambda_k\varphi_k^n\|_{\mathcal{B}}=0.$$We also study the linear combination of composition operators on the Banachalgebra of bounded analytic functions.

Linear combination of composition operators on $H^\infty$ and the Bloch space