Clear[sol0,sol1,sol2,sol3,sol4,sol5,sol6,sol7,sol8,sol9]
Clear[p0,p1,p2,p3,p4,p5,p6,p7,p8,p9]
Clear[λ]
λ=0.5
sol0=p0[t]/.DSolve[{p0'[t]==-λ p0[t],p0[0]==1},p0,t][[1]]
sol1=p1[t]/.DSolve[{p1'[t]==-λ p1[t]+λ sol0,p1[0]==0},p1,t][[1]]
sol2=p2[t]/.DSolve[{p2'[t]==-λ p2[t]+λ sol1,p2[0]==0},p2,t][[1]]
sol3=p3[t]/.DSolve[{p3'[t]==-λ p3[t]+λ sol2,p3[0]==0},p3,t][[1]]
sol4=p4[t]/.DSolve[{p4'[t]==-λ p4[t]+λ sol3,p4[0]==0},p4,t][[1]]
sol5=p5[t]/.DSolve[{p5'[t]==-λ p5[t]+λ sol4,p5[0]==0},p5,t][[1]]
sol6=p6[t]/.DSolve[{p6'[t]==-λ p6[t]+λ sol5,p6[0]==0},p6,t][[1]]
sol7=p7[t]/.DSolve[{p7'[t]==-λ p7[t]+λ sol6,p7[0]==0},p7,t][[1]]
sol8=p8[t]/.DSolve[{p8'[t]==-λ p8[t]+λ sol7,p8[0]==0},p8,t][[1]]
sol9=p9[t]/.DSolve[{p9'[t]==-λ p9[t]+λ sol8,p9[0]==0},p9,t][[1]]
Plot[{sol0,sol1,sol2,sol3,sol4,sol5,sol6,sol7,sol8,sol9},{t,0,50},PlotRange->{0,1}]

summenfkt=sol0+sol1+sol2+sol3+sol4+sol5+sol6+sol7+sol8+sol9
Plot[{summenfkt},{t,0,10},PlotRange->{0,1}]

μ=1
los4=q4[t]/.DSolve[{q4'[t]==- 4μ q4[t],q4[0]==1},q4,t][[1]]
los3=q3[t]/.DSolve[{q3'[t]==-3 μ q3[t]+4 μ los4 ,q3[0]==0},q3,t][[1]]
los2=q2[t]/.DSolve[{q2'[t]==-2 μ q2[t]+3 μ los3,q2[0]==0},q2,t][[1]]
los1=q1[t]/.DSolve[{q1'[t]==- μ q1[t]+2μ los2,q1[0]==0},q1,t][[1]]
los0=1-los1-los2-los3-los4

Plot[{los0,los1,los2,los3,los4},{t,0,10}]

-los0-los1-los2-los3-los4

λλ=Table[i^2,{i,1,10}]

sol0=p0[t]/.DSolve[{p0'[t]==-λλ[[1]] p0[t],p0[0]==1},p0,t][[1]]
sol1=p1[t]/.DSolve[{p1'[t]==-λλ[[2]] p1[t]+λλ[[1]]sol0,p1[0]==0},p1,t][[1]]
sol2=p2[t]/.DSolve[{p2'[t]==-λλ[[3]]p2[t]+λλ[[2]]sol1,p2[0]==0},p2,t][[1]]
sol3=p3[t]/.DSolve[{p3'[t]==-λλ[[4]]p3[t]+λλ[[3]]sol2,p3[0]==0},p3,t][[1]]
sol4=p4[t]/.DSolve[{p4'[t]==-λλ[[5]]p4[t]+λλ[[4]]sol3,p4[0]==0},p4,t][[1]]
sol5=p5[t]/.DSolve[{p5'[t]==-λλ[[6]] p5[t]+λλ[[5]]sol4,p5[0]==0},p5,t][[1]]
sol6=p6[t]/.DSolve[{p6'[t]==-λλ[[7]] p6[t]+λλ[[6]] sol5,p6[0]==0},p6,t][[1]]
sol7=p7[t]/.DSolve[{p7'[t]==-λλ[[8]] p7[t]+λλ[[7]] sol6,p7[0]==0},p7,t][[1]]
sol8=p8[t]/.DSolve[{p8'[t]==-λλ[[9]] p8[t]+λλ[[8]] sol7,p8[0]==0},p8,t][[1]]
sol9=p9[t]/.DSolve[{p9'[t]==-λλ[[10]] p9[t]+λλ[[9]] sol8,p9[0]==0},p9,t][[1]]
Plot[{sol0,sol1,sol2,sol3,sol4,sol5,sol6,sol7,sol8,sol9},{t,0,50},PlotRange->{0,1}]
summenfkt=sol0+sol1+sol2+sol3+sol4+sol5+sol6+sol7+sol8+sol9
Plot[{summenfkt},{t,0,5},PlotRange->{0,1}]

Clear[sol0,sol1,sol2,sol3,sol4,sol5,sol6,sol7,sol8,sol9]
λλ=Table[Log[i+2],{i,1,10}]

sol0=p0[t]/.DSolve[{p0'[t]==-λλ[[1]] p0[t],p0[0]==1},p0,t][[1]];
sol1=p1[t]/.DSolve[{p1'[t]==-λλ[[2]] p1[t]+λλ[[1]]sol0,p1[0]==0},p1,t][[1]];
sol2=p2[t]/.DSolve[{p2'[t]==-λλ[[3]]p2[t]+λλ[[2]]sol1,p2[0]==0},p2,t][[1]];
sol3=p3[t]/.DSolve[{p3'[t]==-λλ[[4]]p3[t]+λλ[[3]]sol2,p3[0]==0},p3,t][[1]];

Plot[{sol0,sol1,sol2,sol3},{t,0,5},PlotRange->{0,1}]
summenfkt=sol0+sol1+sol2+sol3
Plot[{summenfkt},{t,0,5},PlotRange->{0,1}]