Why the impulse response of this system:
and the impulse response of this one:
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First system is a kind of moving average filter (3 samples before + current sample + 3 samples after) with finite memory (7 samples). The impulse response reflects this - it's also finite in time.
Second system is an integrator - the impulse response is a step. It has infinite memory (note that the sum goes from \$k=0\$ to \$+\infty\$), that's why the output is latched to 1 well past the time when you excited it with an unity impulse - an infinite time response.