Question

Demonstrati ca 0,61 < 1 aupra 46 +1 supra 47 + ....+ 1 supra 116 < 1 , 55 . Ex 10

Answer 1

Comparam termenii sumei cu cel mai mare respectiv cel mai mic numar care apartine sumei respective si obtinem:
1/116≤1/116<1/46
1/116<1/115<1/46
1/116<1/114<1/46
......
1/116<1/46≤1/46
Adunand membru cu membru obtinem:
71·1/116<1/46+1/47+...+1/116<71·1/46 ⇔
71/116<1/46+1/47+...+1/116<71/46
Dar 0,61=61/100 unde 61/100<71/116 si
1,55=155/100=31/20 unde 31/20>71/46.
In concluzie am demonstrat inegalitatea de mai sus:
0,61<1/46+1/47+...+1/116<1,55