= Solution
Take the <Kronecker quiver>, with two arrows $\alpha,\beta:1\to2$. There are no paths of length two, so its <path algebra> has basis $e_1,e_2,\alpha,\beta$ and dimension four.
For each $\lambda\in k$, take a <quiver representation> with one-dimensional spaces at both vertices and arrow maps $(1,\lambda)$. An endomorphism consists of scalars $a,b$ with $b=a$ from the first arrow; the second imposes no further condition. Its <endomorphism ring> is therefore $k$, which has no nontrivial <idempotent>. Thus the representation is indecomposable. An isomorphism to the representation $(1,\mu)$ again requires $b=a\ne0$, and the second arrow then gives $\lambda=\mu$. Because an <algebraically closed field> is infinite, this gives infinitely many pairwise nonisomorphic indecomposables. Hence
$$
\boxed{\dim_k kQ_2=4,\qquad kQ_2\text{ is not of finite representation type}.}
$$
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