1 (* ========================================================================= *)
2 (* Sum of reciprocals of triangular numbers. *)
3 (* ========================================================================= *)
5 needs "Multivariate/misc.ml";; (*** Just for REAL_ARCH_INV! ***)
9 (* ------------------------------------------------------------------------- *)
10 (* Definition of triangular numbers. *)
11 (* ------------------------------------------------------------------------- *)
13 let triangle = new_definition
14 `triangle n = (n * (n + 1)) DIV 2`;;
16 (* ------------------------------------------------------------------------- *)
17 (* Mapping them into the reals: division is exact. *)
18 (* ------------------------------------------------------------------------- *)
20 let REAL_TRIANGLE = prove
21 (`&(triangle n) = (&n * (&n + &1)) / &2`,
22 MATCH_MP_TAC(REAL_ARITH `&2 * x = y ==> x = y / &2`) THEN
23 REWRITE_TAC[triangle; REAL_OF_NUM_MUL; REAL_OF_NUM_ADD; REAL_OF_NUM_EQ] THEN
24 SUBGOAL_THEN `EVEN(n * (n + 1))` MP_TAC THENL
25 [REWRITE_TAC[EVEN_MULT; EVEN_ADD; ARITH] THEN CONV_TAC TAUT;
26 REWRITE_TAC[EVEN_EXISTS] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
27 AP_TERM_TAC THEN MATCH_MP_TAC DIV_MULT THEN REWRITE_TAC[ARITH]]);;
29 (* ------------------------------------------------------------------------- *)
30 (* Sum of a finite number of terms. *)
31 (* ------------------------------------------------------------------------- *)
33 let TRIANGLE_FINITE_SUM = prove
34 (`!n. sum(1..n) (\k. &1 / &(triangle k)) = &2 - &2 / (&n + &1)`,
36 ASM_REWRITE_TAC[SUM_CLAUSES_NUMSEG; ARITH_EQ; ARITH_RULE `1 <= SUC n`] THEN
37 CONV_TAC REAL_RAT_REDUCE_CONV THEN
38 REWRITE_TAC[REAL_TRIANGLE; GSYM REAL_OF_NUM_SUC] THEN CONV_TAC REAL_FIELD);;
40 (* ------------------------------------------------------------------------- *)
42 (* ------------------------------------------------------------------------- *)
44 let TRIANGLE_CONVERGES = prove
47 ==> abs(sum(1..n) (\k. &1 / &(triangle k)) - &2) < e`,
48 GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN
49 DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN
50 EXISTS_TAC `2 * N + 1` THEN REWRITE_TAC[GE] THEN REPEAT STRIP_TAC THEN
51 REWRITE_TAC[TRIANGLE_FINITE_SUM; REAL_ARITH `abs(x - y - x) = abs y`] THEN
52 MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N)` THEN
53 ASM_SIMP_TAC[REAL_ABS_DIV; REAL_ABS_NUM] THEN
54 ONCE_REWRITE_TAC[GSYM REAL_INV_DIV] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
55 SIMP_TAC[REAL_LE_RDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
56 REWRITE_TAC[REAL_ARITH `abs(&n + &1) = &n + &1`] THEN
57 REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_MUL; REAL_OF_NUM_LE] THEN
58 POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN ARITH_TAC);;
60 (* ------------------------------------------------------------------------- *)
61 (* In terms of limits. *)
62 (* ------------------------------------------------------------------------- *)
64 needs "Library/analysis.ml";;
66 override_interface ("-->",`(tends_num_real)`);;
68 let TRIANGLE_CONVERGES' = prove
69 (`(\n. sum(1..n) (\k. &1 / &(triangle k))) --> &2`,
70 REWRITE_TAC[SEQ; TRIANGLE_CONVERGES]);;